Martin plunged to his left to keep out a fierce 12-yard shot from Jake Robinson just seconds before the referee blew the final whistle.
Vale were the brighter side from the off and Paul Marshall's 17th-minute shot would have found the net if it wasn't for a timely block by Dean Howell.
Aldershot relied more on the long ball as they forced Martin to race from goal to clear his lines on a couple of occasions.
But Vale continued to hold the upper-hand and it needed a fine save by Nikki Bull, diving low to his left, to keep out a curling effort from Louis Dodds.
And Dodds drew another impressive stop from Bull, this time high to his left, as a largely uneventful first half drew towards a close.
Aldershot, whose passing had been way off-beam, were fortunate to remain on level terms as they trooped off for the break.
Dodds was again in the thick of the action at the start of the second period when his powerful shot from distance warmed Bull's fingers.
The visitors did show signs of improvement as Danny Hylton fluffed a half-chance before the pace of Robinson almost found a way through the Vale rearguard.
But Vale twice came desperately close to taking the lead following James Lawrie's low cross in the 71st minute.
Rob Taylor should have scored when his initial shot was blocked by Lewis Chalmers and then Paul Marshall smashed the follow-up against the crossbar.
The contest was finally livening up as Shots substitute Rob Elvins took too long to shoot after Louis Soares' pass had sent him clear.
And Dodds again went close eight minutes from time when he guided his header inches wide from Paul Edwards' cross, this time with Bull well-beaten.