The result means that Harps are still six points adrift of the Tribesmen who occupy third spot.
Harps began brightly but Galway had the best chances in the first half.
The game opened up in the second half and both sides had chances to win it in the final 20 minutes.
On three occasions Sean McCarron had opportunities for Harps - and he even hit the crossbar - while Jake Keegan gave Harps later scare with a header from a good position.
In the end a draw was a fair result.
Commenting on the game, Harps boss Ollie Horgan spoke with Diarmuid Doherty