Well, then, let me goof on Elvis to mention that “Viva Las Vegas” of course references blackjack and poker and the roulette wheel along with shooting dice.
Whatever happened to Pong - Frank Black
Computer games - Mi-Sex - 1980s Australian band
The Joker - Anthony Newley [fantastic original] - Kimberly Diane “Kim” Craig (née Day) [heaps better]
Trudy by the Charlie Daniels Band. Another poker game gone wrong song.
Snake Eyes - The Alan Parsons Project
Shooting dice.
Stretching it a bit:
Many a tear has to fall
But it’s all
In the game
All in the wonderful game
That we know
As love …
Come on baby, let’s start to play,
Come on baby, let’s play
The game of love (love), love (love), lala lala lala …
Hit Somebody! (The Hockey Song)–Warren Zevon
Glory Days–Bruce Springsteen
Boom Boom Mancini-- Warren Zevon
I know it’s sports, but Sister Wynona Carr’s "Life is a Ballgame" is too much fun to ignore.
*… The first base is temptation
You know the second base is sin
The third base is tribulation
If you pass, you can make it in
Old man Solomon is the umpire
And Satan is pitchin’ a game
He’ll do his best to strike you out
He playin’ just the same …*
Sorry, didn’t catch the end of the OP about sports.
No specific game is mentioned, but the first song I thought of is “Games People Play” by Alan Parsons Project.
https://www.youtube.com/watch?v=SLi7Ljcy6n8
“Game On,” a Bollywood-style song about video games from “The Guild.”
“Roll a D6,” about playing Dungeons and Dragons.
You’re right, I misremembered it. My first exposure to the Stagger Lee myth was the Clash’s Wrong 'Em Boyo", which starts with the lines:
*Stagger Lee met Billy
And they got down to gambling
Stagger Lee throwed seven
Billy said that he throwed eight
So Billy said, “Hey Stagger
I’m gonna make my big attack
I’m gonna have to leave my knife
In your back”
*
So they were playing dice, obviously.
Hand Me The Wine and The Dice from Aspects of Love sort of qualifies.
OTOH, I cannot think of a song that qualifies more than Tom Paxton’s The Name of The Game is Stud
Oh Lordy, don’t you see
This old Stud Poker be the ruin of me
You got to get ready to shed some blood
When the name of the game is Stud
Mr. Mudd and Mr. Gold– Townes Van Zandt.
I’m listening to AC/DC’s “Bonfire Box”. How could I forget about “The Jack”. Or isn’t it about a card game? 
Or for a super brief reference Rod can “steal my daddy’s cue” in “Maggie May”.
A 1991 Barry Manilow number mentions scrabble, but it was surrounded by such intriguing lyrics that I thought it warranted a good sample:
*The books we can split in half
The pots and pan we’ll do the same
The pictures on the wall
We can split in half
And you can have the scrabble game
The money in the bank
We can split in half
And you can have your single name
Our friends, we can split in half
And we can each have half the lane
OK it’s settled then
You can have the t.v.
I’ll take the radio
Really I don’t want the t.v.
Just please the radio
I’ll like a lamp if you don’t want it
Thank you
You keep the couch you’ll need a couch
That’s OK
You keep the bed, I’ll buy a bed
That’s OK
The dog, you keep the dog
I’ll come around and see him now and then. *
end of sort of derail.
Steve Goodman’s City of New Orleans:
Dealing card games, with the old men in the club car
Penny a point ain’t no one keeping score
Pass the paper bag that holds the bottle
Feel the wheels, rumblin’ 'neath the floor
Shirley Ellis’s The Name Game, where the song is the game.
Probably stretching the definition of “game” too far, but The Mountain Goats put out the album “Beat the Champ” in 2014, a concept album with multiple songs about professional wrestling. It’s quite good.