Belgian striker Benteke, who last week signed a new four-year contract at Villa Park, hit three goals in 13 first-half minutes after Andreas Weimann had opened the scoring from distance after two minutes of action at Gresty Road.
Benteke grabbed his first from eight yards after being played in by Weimann, and then latched on a clever pass from Karim El Ahmadi to finish powerfuly in the 28th minute.
He completed his treble six minutes later after cutting in from the left and firing his shot across the keeper.
Harry Davis pulled a goal back for Crewe just before the break and Villa manager Paul Lambert decided to make 10 changes at half-time.
The Premier League side lost at Luton on Tuesday after being held to successive draws at Wycombe, Bochum, Paderborn and Wigan but there was no stopping them in Cheshire and Dutch winger Leandro Bacuna added a fifth goal seven minutes from time.