Despite a goalless first half, both sides produced some high-quality football. Neither side hinted at their non-league past and the visitors' display marked them out as the strongest side to have visited the EBB Stadium this season.
In the opening exchanges Marvin Morgan's header from six yards was gratefully received by Paul Jones after deflecting off Tully.
On seven minutes Steve Basham header was inches over the bar after he rose highest to meet Dean Moxey's corner.
Quarter of the way through the contest Morgan's 20-yard drive was collected easily by Jones. Exeter countered and in the ensuing goalmouth scramble Marcus Stewart betrayed his top-flight credentials, heading high and wide when the ball fell kindly to him.
With three minutes remaining Kirk Hudson broke from his own half and found Morgan inside the box, but the striker headed over the bar.
Slightly against the run of play Grant put Aldershot ahead on 52 minutes. Tully failed to deal with Scott Davies' long ball and Grant stole in to score his second goal in a week.
Moments later Moxey cut on to his right foot on the edge of the area, his drive forcing an excellent diving save from Nikki Bull.
Just before the hour mark, the lively Adam Stansfield was denied a certain goal by Anthony Charles' last-ditch tackle in the box.
Exeter piled on the pressure, on 67 minutes Stansfield misfired a volley from five yards from Moxey's cross.
Minutes later, substitute Manny Panther shot weakly at Bull after a skilful run into the box.
With a quarter of an hour to go, Grant had chance to finish the game when Moxey shoved Davies in the penalty box.
From the spot Grant shot powerfully, but Jones produced a stunning double save to keep Exeter in the game.
As time ran out Ben Watson had a great chance to equalise, but blazed horribly high and wide from five yards out.