Akinfenwa netted the opener on 26 minutes when he cushioned a side-foot volley into the bottom corner to continue his impressive scoring run.
Lowe almost levelled the score up moments after the goal when he struck the bar with a free kick, before Akinfenwa missed a chance to double Wimbledon's lead with a close-range effort.
However, the forward was on hand to net his seventh goal in six games when he converted with a powerful header on 37 minutes.
The visitors reduced the deficit four minutes after the restart through Mayor, but Tubbs restored Wimbledon's two-goal advantage on 53 minutes.
Bury tried to force a late equaliser late on, but the home side stood firm to hold on for the victory.