Rashford had just been booked for bringing down Matty Cash and then proceeded to foul Leon Bailey.
The 26-year-old managed to escape without a second booking and was promptly substituted by Erik ten Hag, with Antony replacing him.
Carragher was adamant that Rashford should have been dismissed.
Speaking on Sky Sports, Carragher said: 'If Rashford hadn't already got a yellow card, that is a yellow. It should have been a red card and Manchester United should be down to ten men.'
Jamie Carragher claimed Marcus Rashford should have been sent off against Aston Villa
Rashford had previously been substituted at half time in United's draw with Porto in the Europa League in midweek despite having scored earlier in the game.
So far this campaign Rashford has scored four goals in eleven appearances for United.
The Rashford incident aside, it was a largely uneventful encounter at Villa Park, with the game ending 0-0.
However, ten Hag will be pleased with an improved defensive performance from his side.