Adams, who began his career at Gigg Lane, clinched it for the Reds with his fifth goal in four games after the bottom-of-the-table Shakers had staged a stirring second-half fightback to get level at 2-2.
Crawley looked in control after goals from Adams and Hope Akpan put them 2-0 ahead nine minutes into the second half. But Bury scored twice in the space of three minutes through Tom Hopper and Andy Bishop to put the match on a knife-edge.
Barker said: "At 2-0 up we were looking comfortable but then we switched off for a throw-in and Bury were back in the game.
"Fortunately we found an extra gear and I'm sure the match was an entertaining one to watch, but if you're not at your best for 30 per cent of the game, you'll get punished as you're playing against good players.
"Most of the time it was a good performance but it may have cost us if we were up against a better strikeforce. You won't dominate a game at this level for 100 per cent of the time."