League One side Dons arrived at Barnet having won their last five games and headed into the half-time break a goal to the good after Darren Potter drove home.
Charlie MacDonald made it two with 10 minutes to go, but just when it looked like Barnet would make for a nervous final few minutes after Izale McLeod had found the net, Powell went up the other end to quell any hopes the Bees had of forcing a replay.
Barnet goalkeeper Dean Brill kept his side in the game with some great saves, but was helpless in the 39th minute as Potter found the bottom left corner from 18 yards.
The visitors continued to press, but were made to wait until the 78th minute for their second, MacDonald rounding Brill before passing into an unguarded net.
McLeod gave Barnet hope when he smashed home the rebound from a Sam Deering effort, but Dean Lewington released Powell, who kept his cool with a one-on-one with Brill to settle the tie.