Edwards bundled home from close range after Tranmere goalkeeper Danny Coyne had dropped Matthew Spring's cross into his path.
The win gave administration-hit Hatters a vital three points in the battle to beat the drop in League One.
With both sides playing five across the middle there was little opportunity for invention as the game got off to a slow start.
The first incident of note didn't arrive until the 17th minute and that was an enforced substitution rather than on-the-field action, as Paul Furlong limped off to be replaced by Calvin Andrew.
Darren Currie flashed a 22-yard effort just past the post for Luton in the 24th minute and Shane Sherriff repeated the trick for Tranmere three minutes later with a 15-yard volley that flew off target.
But excitement was at a premium and the first shot on target took 32 minutes to arrive when Currie's 30-yarder forced Coyne to collect at the second attempt.
A turgid first half ended with Chris Greenacre and Gareth Taylor both being booked as the game struggled to ignite.
The second half continued in much the same pattern with only a Gareth Taylor header at Dean Brill in the 50th minute and a wayward 20-yard effort from Edwards six minutes later lifting the gloom.
The Hatters eventually moved to two up front as Drew Talbot replaced Currie in the 67th minute.
And the move had an instant impact as two minutes later Talbot fed Spring on the left and, from the midfielder's cross, Coyne dropped the ball at the feet of the onrushing Edwards, who poked home from a yard out.
Tranmere should have equalised a minute later as Chris Shuker broke down the left and pulled the ball back for Andrew Taylor, only for the left-back to sidefoot wide with the goal at his mercy.
Edwards then missed the chance to double the hosts' lead as he fluffed a six-yard header from David Bell's 72nd-minute cross.
Brill preserved Luton's lead with a fine block from Gareth Taylor seven minutes from time.