It moves Annan level on points with the Bully Wee in the battle for a promotion play-off spot.
Cowdenbeath moved off the bottom of the table after Lewis Moore's second-half goal secured a 2-1 victory away to Stirling Albion.
Berwick Rangers slipped to the foot of the table after falling to a 3-1 defeat at home to in-form Edinburgh City.
Peter MacDonald opened the scoring for fourth-placed Clyde after 12 minutes before two goals in 11 second-half minutes from David McKenna put Annan in front.
MacDonald equalised from the penalty spot with 10 minutes left, but Smith fired home the winner two minutes from the end of normal time.
Kris Renton gave Cowdenbeath the lead against Stirling just after the break before Dylan Nguene Bikey levelled for the Binos after 54 minutes.
However, on-loan Hearts midfielder Moore struck 10 minutes later for the Blue Brazil, who ended a run of three straight defeats.
Two goals in the opening three minutes, scored by Chris McKee and Douglas Gair, helped Edinburgh on their way against Berwick at Shielfield Park.
Craig Beattie added a third with a penalty after 71 minutes before Jordyn Sheerin's consolation moments later.
Edinburgh are now unbeaten in the league for eight games.
Aidan Smith's late goal secured Annan Athletic a 3-2 victory away to Clyde in League Two.