Two stoppage-time goals from Annan gave Montrose a post-Christmas hangover and saw the Borders side climb to third in League Two.
Montrose had a scare in six minutes when Alan Campbell had to clear an effort from Kenny McKay off the line.
Montrose hit back, and in 11 minutes an effort from Garry Wood was saved by Kenny Arthur, then, in 19 minutes a chip from Paul Watson was saved.
Craig Bell had the ball in the net for the home side in 26 minutes, but was given offside.
Graham Webster headed wide from a Watson free-kick, then Watson headed wide from a Terry Masson cross.
Five minutes into the second half a drive from Wood was tipped over by Arthur.
The teams continued to cancel each other out and a 0-0 draw seemed the likely outcome, then, as the match moved into stoppage-time, Annan’s Jordan Moffat broke on the right and his cutback was fired home by McKay.
Three minutes into added time, McKay crossed from the byline and Jack Todd hooked home from close range.
Montrose boss Stuart Garden said: “I am disappointed to lose right at the end as we didn’t deserve to.”
He continued: “The goals we conceded were very soft. A couple of bad decisions defensively cost us.
“The prize for the team that won the game was third place, but it wasn’t to be for us.”