The phrase game, set (and) match denotes:
– (literally) a victory in a tennis match, secured by winning the deciding game of the last set required to win;
– (in extended use) a complete and decisive victory.

This phrase is, for example, punned upon in the following advertisement for the British retailer Marks & Spencer, published in The Guardian (London and Manchester, England) of Saturday 15^th July 2023:

Game, set…
match(ing) knickers
M & S

The earliest literal use of the phrase game, set (and) match that I have found is from the account of a tennis match between Edmund and John Tompkins and George and William Lambert, which took place on Tuesday 23^rd May 1876 at Lord’s tennis court, London, published in The Field, The Farm, The Garden, The Country Gentleman’s Newspaper (London, England) of Saturday 27^th May 1876:

The Lamberts gained another stroke, but Edmund won the score back to deuce; the next stroke he lost again, and then followed a splendid rest, which was at length decided by G. Lambert winning the game, set, and match; 3 sets love.

The earliest extended use of the phrase game, set (and) match that I have found is from the column This Busy World, by ‘Busy B.’, published in The Manchester Weekly Times and Salford Weekly News (Manchester, Lancashire, England) of Saturday 14^th July 1906:

It isn’t always policy to interfere with a woman’s tastes, Even a mill-girl will turn, as a Blackburn vicar has recently discovered. The parson made a few remarks about the alleged extravagance of our Lancashire lasses who toil and spin, and spin jolly hard, while some people are lounging in their studies and dawdling over the teacups of fair parishioners.
The vicar will be thinking he caught a Shrewfragette, for a mill girl has tartly retorted in the following chili-paste and Nabob pickles manner:
Extravagant indeed! I should like to know where it comes in. From six in the morning until nearly that hour in the evening we live in an atmosphere which neither parsons nor their wives would tolerate for many minutes.
A palpable hit, methinks! And this, too, is constructed so as to resemble a neat polemical upper-cut.
And the vicar says something about “latest Paris or Bond-street fashions,” as if we are not entitled to wear the latest styles as anyone else. I can tell you this: What we do buy we pay for, and that can’t be said by a lot of the “top nobs.” I think, as a Solomonesque umpire, I may call “game, set, and match” for the lady.

The phrase game, set (and) match came to be used in 1922 in a British game called Beaver. The first two occurrences that I have found are as follows:

1-: From a letter to the Editor, by ‘Rufus Hirsute’, published in the Gloucestershire Echo (Cheltenham, Gloucestershire, England) of Friday 8^th September 1922:

Sir,—l shall be grateful if you can see your way to make space in your columns for the following protest:—
I am the owner of a green bicycle. I also wear a beard, and my hair is of an auburn tint. The result of this combination is that I dare scarcely move from the privacy of my apartments except by night, and then only at the risk of what must be described as continued and studied insult. You ask me “Why?” The answer is, sir, that I am, in the words of the ribald, a beaver. More than that I am, it seems, a “King” beaver. Whenever I venture forth I am assailed on all sides by wild cries of “thirty,” “deuce,” “game set and match,” and all the other strident technicalities of this cruel and debasing sport.

2-: From To-Day’s Tonic Talk. Are You Mentally Agile?, by Dr. Frank Crane *, published in the Pall Mall Gazette and Globe (London, England) of Wednesday 27^th September 1922:

“Beaver,” recently introduced by British University students, is likely to spread to the farthest corners of the Anglo-Saxon world. It is a “silly” game, but as the “New Statesman” points out in an unsigned editorial, “It is a silly game in the English tradition of silliness.”
And that makes a difference.
“Beaver” is played by two persons, and the points are scored as in tennis. The object of the game is to find beards on the faces of the male population around you.
Whichever of the two players first cries “Beaver!” as a beard heaves into sight scores. At sight of a white beard one cries “Polar Beaver!” which counts as a game. At sight of a Royal beard the proper call is “Royal Beaver!” which counts, not only as a game, but as set and match. There is a story—of course, untrue—of a Cambridge function, at which, on the entrance of a Royal figure wearing a beard, the entire audience of undergraduates rose to their feet with a shout of “Royal Beaver! Game, set, match!”

[* Probably the U.S. Presbyterian minister Dr. Frank Crane (1861-1928).]

The other early extended uses of the phrase game, set (and) match that I have found are as follows, in chronological order:

1-: From this advertisement for Barclay’s London Lager, published in The Tatler (London, England) of Wednesday 6^th June 1928:

THE “LIGHT OR DARK” WAY TO QUENCH EVERY THIRST

’VANTAGE IN!
Game—set—match . . . Barclay’s!

2-: From Ritz—Carlton, an obscure dialogue published in The Bystander (London, England) of Wednesday 26^th December 1928:

“Morning, Ritz.”
“Morning, Carlton.”
“See that the New York papers made a set at Lord Inverclyde when he arrived?”
“Yes; what’s their game?”
“It looks like a case of game, set, love—match to me.”
“My dear Carlton, for goodness’ sake don’t be so je-june. Still, I’ll admit that the Lido’s a wonderful place.”

3 & 4-: From the column Day by Day, published in the Evening Telegraph and Post (Dundee, Angus, Scotland):

3-: Of Tuesday 9^th April 1929:

A telephone subscriber complains that a girl at the Exchange gave him a four-figure number in which not one figure was right. That counts game, set, and match to her.

4-: Of Monday 12^th August 1929:

“Lady ——,” writes a gossip, “always looks to me exactly right.” Game, set and match to her.