The Shakers struggled to get on top of the Blue Square Premier side at Gigg Lane, but sustained pressure eventually yielded Adam's eighth goal of the season - his fifth in the last seven games.
Exeter deserved more from the game, but for two superb saves from home keeper Jim Provett, could have easily been looking at a replay at St James' Park a week on Tuesday.
The first chance of the game fell to Andy Bishop in the sixth minute, the striker saw a powerful right-foot shot deflected wide for a corner.
At the other end Exeter nearly caught the home side cold with a quick counter-attack, but from a useful knock-on from Jamie Mackie, Steve Basham fired over the bar.
A few minutes later Matt Gill's dangerous corner from the left wing should have been headed home by the unmarked Jon Richardson, but the big defender sliced his header horribly wide.
Bury broke and Andy Mangan wasted a good opportunity when he failed to hit the target from 12 yards out.
Exeter, 21 places below their opponents in the league standings, didn't look out of their depth, but a sweeping move in the 34th minute should have seen the hosts go ahead.
A delightful long ball from Colin Woodthorpe found Bishop wide on the left wing. His pass across goal went behind Adams and there was David Buchanan in acres of space.
However, the midfielder struck his first-time left-foot shot straight at Paul Jones and the keeper made a smart close-range stop to deny him.
Mackie found a yard of space in the 38th minute, but had his right-foot shot charged down on the edge of the penalty area.
Mangan, who had a first-half to forget, had a good chance in the 54th minute, but from 20 yards out dragged a left-foot shot well wide of the target.
It came as no surprise when Mangan was replaced on the hour mark and substitute Liam Hughes was given the chance to shine.
Exeter skipper Gill tried his luck from distance in the 63rd minute, but his right-foot curling shot was easy for keeper Provett.
Moments later Hughes nearly gave Bury the lead, but his rasping left-foot drive flew into the sidenetting.
The home side had Provett to thank for keeping them on level terms in the 71st minute. The keeper produced an excellent save to turn Wayne Carlisle's strong effort past the left-hand post.
A clever backheel from Bishop set Adams through on goal in the 76th minute, but the youngster was denied a goal after a wonderful save from Jones, the keeper turning the striker's sweet left-foot strike over the bar from close range.
But the breakthrough eventually came when Exeter failed to clear a corner from the right, Woodthorpe's left-foot drive was parried by the otherwise impressive Jones and there was Adams to fire home the rebound.
Bishop should have done better with a short-range header in the 85th minute from a Buchanan corner and a minute later forced Jones to make another stop.
However, substitute Richard Logan then forced Provett to pull-off a brilliant save from a header on the edge of the six-yard box.
Bishop had another good chance with two minutes left, from dragged a left-foot shot wide of the far post having held off the challenge of two defenders.