What was the first tv show that successfully continued after losing a co-star from an established pair of characters?

There's a case to be made for Bewitched (1964 - 1972), which continued for three seasons after the co-star Dick York was replaced by Dick Sargent in the same role of Darrin Stephens. The main character, Samantha Stephens, was married to Darrin, so they definitely qualify as a couple/duo.

Watching syndicated re-runs of Bewitched as a kid, I actually didn't even notice there were two different actors playing Darrin.

Yes it has happened.

An early example is Doctor Who. Every few seasons either The Doctor regenerates and begins to be played by a new actor, or he switches his companion and gets a new one, again played by a new actor. It has been running for ages and is still successful.

As @KutuluMike noted, the success or failure of losing actors or getting new ones is determined by the strength of the replacement character. In Doctor Who, the character does not need to be replaced, because he merely regenerates, with an entirely new body that can be played by a different actor.

_{(Historical note: This answer was written before the question was edited to ask for "the first". I'm not changing it because I think it's more informative the way it is.)}

Yes, it has happened, though it's rare. The trick seems to be how strong of a replacement character they can get to fill the void, and how well the show's supporting cast is at holding it together in the mean time.

A good example of a success would be Cheers. Though it was technically an ensemble show, for all intents and purposes, the stars of Cheers were Ted Danson and Shelley Long. Long left in the fifth season and was replaced by Kirstie Alley and the show would continue to be successful for another six seasons. This is largely due to the fact that Danson and Alley also developed a good on-screen chemistry, while the rest of the cast was strong enough to keep the show around while that happened.

Two And A Half Men is another example, though one could argue that losing Charlie Sheen and replacing him with Ashton Kutcher doesn't count as "successfully continuing"; the show did OK for the first two seasons post-Sheen but plummeted in the ratings after that. Again, the presence of Angus T. Jones is likely what help keep the show around (after he left, the show did tank in the ratings and get cancelled).

There is also a fairly large list of cases where a show loses it's lead actor, but the character s/he plays remains on the show. This is less common in modern TV than is used to be, and usually happens to bit parts or recurring characters, but it has been done with lead roles. The two "trope namers" (which may not be the earliest examples, but are by far the most well-known) are Bewitched, which swapped out Dick York for Dick Sargent in the lead role of Darrin, and Doctor Who, which made replacement of the actor playing The Doctor a key plot element from very early in the show's run.

(Historically speaking, by the way, this does not bode well for Castle, but may mean Sleepy Hollow has a chance. Of course, both are doing terrible with both co-leads, so it may be irrelevant.)

The Goldbergs is possibly the earliest example, running from 1949 to 1957, originally starring Phillip Loeb as Jake Goldberg, but the Red Scare forced him off the show and he was replaced by Harold Stone and Robert H. Harris (sources differ as to which was first).

Phillip Loeb, blacklisted on suspicion of being a communist, eventually committed suicide.

Edit: TVGuide.com seems to give a more definite date listing for the cast. Phillip Loeb starred from 1949-1951. Harold J. Stone took over in 1952, and Robert H. Harris replaced him from 1953 onward.

The George Burns and Gracie Allen Show aired October 12, 1950 – September 15, 1958, continued for about 5 years after the last of 3 switches by the actor playing neighbour Harry Morton.

Harry Morton was first portrayed by Hal March (October–December 1950), then John Brown (January–June 1951), and after that, Fred Clark, until 1953 when the role was assumed by Larry Keating.

It's a funny story where the last switch was introduced:

In one famous episode, "Morton Buys Iron Deer/Gracie Thinks George Needs Glasses", George walks on-stage and freezes the scene just before Harry's entrance and explains that Clark has left the show to perform on Broadway. He introduces Larry Keating, who enters, and then calls over Bea Benaderet to introduce the two saying, "This is Larry Keating and he is going to be your husband now". The pair greet and chat briefly, complimenting each other on their previous work. George remarks that if they are going to be so nice to each other, no one will believe they are married. Burns then gives a cue, Blanche resumes her position, and the scene continues where it stopped as if nothing had happened. The new Harry enters and Blanche hits him in the head with a catalog for spending $200 to buy an iron deer.

The entire episode is on YouTube, and George's "stop the camera" bit starts at 8:38.

_{This is probably the 2nd oldest answer so far, The Goldbergs('49-'57) switch apparently took place after it was dropped from the schedule in June 1951 and picked up for the 1952-53 season}

While it was a sitcom with three main characters rather than a duo, and post-dates the other examples given, the show Game On successfully replaced a Hollywood-bound Ben Chaplin with Neil Stuke after the first season.

It's worth a mention if only for the nice way they addressed the casting change in-show:

It also happened in the show Big Bad Beetleborgs when a young actress was replaced. In the show it was explianed as a transformation from a villain character and in order to hide the transformation from her friends and family she was given a totem (I believe a necklace) that would allow everyone around her to see her original body.

Now that show was about a trio more than a duo but the change didn't really affect the show's performance. (That said I think burnout of the genre and low overall quality did)

One early show that hasn't yet been mentioned is Make Room for Daddy. Co-star Jean Hagen quit the series after three seasons, so her character was killed off (a television first!) and the series continued under the new title The Danny Thomas Show for eight more seasons. Though it wasn't the first to continue without a co-star, it was probably the first to do so without recasting the character, and the most successful in terms of how long the series continued after the change.

The Avengers (UK series 1961-9)

Created as a star vehicle for Ian Hendry, who played Dr David Keel, who fought crime assisted by his enigmatic partner, John Steed.

After the first season, Hendry quit. Steed was promoted to lead character, assisted by Cathy Gale, then Emma Peel, then Tara King. (And a couple of forgettable short-timers)

Not affecting a "pair", but Bugs was a program from the 90's in the UK that ran for four series with four main characters. At the end of the third series the actor playing one of the main characters (Ed) changed. It went from the Australian Craig McLachlan to the Yorkshire-born Steven Houghton. As far as I recall the casting change was not referenced in the show in any way.

The show didn't carry on after the 4th series, but it's not clear whether this was due to the change in actors (as it nearly didn't get renewed after the 3rd series)