Steven Gerrard's side had to come from behind to avoid an upset against League One Trotters.
Dion Charles' 24th-minute opener gave the visitors an early lead.
Five-time winners Villa were level before the half-time break thanks to a goal from Douglas Luiz.
Lucas Digne and Leon Bailey finished off the victory.
Following Sunday's disappointing 3-1 loss to Crystal Palace, Gerrard said his side would treat Tuesday's tie with seriousness.
Pressure is threatening to build on the former Rangers boss after a run of three league wins in 14 games since March and the Villa boss named a strong starting line-up.
Philippe Coutinho and Danny Ings were among five players to come in, though it was a determinedBolton who fired the first warning shot.
The stadium erupted after Charles beat Calum Chambers after excellent work by Kieran Lee.
It was a strange goal as the set-piece evaded the reach of the goalkeeper and flew into the top corner.
Ings clipped the post from an angle after the restart as the visitors looked a lot brighter after the break.
The forward was marginally onside just after the hour as he raced through on goal before being felled by Dixon's outstretched leg.
Three minutes later, Digne extended the advantage, while a superb individual goal from Bailey, who cut in from the right before finding the far bottom corner, capped a much improved second half display.
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