The home side should have taken the lead in the 10th minute when Padraig Amond latched on to a deflected cross at the back post, but he could not hit the target from close range.
Grimsby threatened again when Danny East collected the ball in the box and beat two defenders to rifle a strike at goal, but Dillon Phillips made the save.
Paul Hurst's men should have taken the lead as Omar Bogle slipped a through-ball into the path of Amond, who only had Phillips to beat, but the keeper made a fine stop with his body.
Cheltenham tested McKeown just before the break when they won a free kick on the edge of the box, but the keeper made a good stop to claim Kyle Storer's curling effort.
Cheltenham took the lead in the 68th minute when Munns lifted a corner in from the right and the hosts failed to clear the danger, allowing Downes to turn and drill an effort into the top corner.