Defeat leaves bottom-of-the-table Pools without a win or a goal in seven outings and they are eight points adrift of safety with five games to go.
The struggling hosts were reduced to 10 men on 51 minutes when Simon Walton was red carded after leaving a foot in on Patrick Bamford.
The Dons midfielder opened the scoring on just six minutes. After Pools lost possession, he waltzed through the home defence to confidently roll in a low finish in from 10 yards.
Minutes after Walton's red card, Bamford could have been sent-off for a needless push on Jack Baldwin, but was booked.
And the Dons completed their first away win since January 1 after 62 minutes.
Dean Bowditch curled home a low finish from 18 yards following a swift counter-attack.
Pools' best chance came on 84 minutes when skipper Peter Hartley fired across goal against the far post with goalkeeper Ian McLoughlin beaten.