Villa led 2-1 at halftime and Benteke scored after 55, 59 and 72 minutes to lift them to up to 16th place in the table on 37 points with three matches left, five points ahead of third-bottom Wigan Athletic who have a game in hand.
Sunderland, who had Stephane Sessegnon sent off after 70 minutes, and Newcastle United also have 37 points from 35 games.
Villa captain Ron Vlaar broke the deadlock after 31 minutes, rifling a low shot from 30 yards into the corner of the net.
The home side's lead lasted less than a minute, however, as Sunderland defender Danny Rose played a neat one-two with Danny Graham before slotting a composed finish past Brad Guzan.
Villa were stunned but they regained the lead six minutes later when Matthew Lowton's cross was calmly finished from 12 yards by forward Andreas Weimann.
Benteke grabbed his first by heading in a rebound and he struck again when he powered home a header from Ashley Westwood's corner.
The Belgian striker completed his hat-trick with a cool angled finish and Gabriel Agbonlahor completed the rout two minutes from time.