With the match nicely poised at 2-2 going into the closing 16 minutes, Peat raced out of his penalty area to clear but missed the ball completely, allowing David McKenna to gather and find the empty net.
A difficult task soon became an impossible one when visiting midfielder Paul Watson was red-carded for an off-the-ball kick at Liam Buchanan.
That man Peat was at it again two minutes from time as he failed to deal with a routine corner kick, and McKenna accepted his second gift of the afternoon with a fierce angled finish.
Armand One had earlier fired Cowdenbeath ahead with a low drive from 16 yards, but Gavin Swankie equalised with a perfectly-judged volley after 25 minutes.
Kevin McMullan then rattled the home woodwork with a superb curling effort from long range, and he finally got his reward early in the second half when he danced past a series of weak tackles before beating keeper David Hay with his left foot from the edge of the penalty box.
But Arbroath's lead lasted only 60 seconds as top scorer Buchanan reacted quickest inside a crowded six-yard area to hammer home his 14th goal of the season.