Marc Albrighton gave the Premier League visitors a first-half lead before Gabriel Agbonlahor and Ciaran Clark struck after the break to put them in complete control.
Rovers, with caretaker boss Shaun North in charge following the midweek departure of Paul Buckle, battled gamely and pulled one back in added time through Scott McGleish, but the veteran then missed the chance to set up a thrilling finale as his penalty was saved by Brad Guzan.
While his side are in the draw for the next round, Villa boss Alex McLeish has injury worries over Emile Heskey and James Collins who were forced off.
McLeish had named a strong side but, with a passionate full house behind them, the League Two strugglers, who lie 74 places below Villa in the league pyramid, made an encouraging start in their hunt for an upset.
Chris Zebroski raced past Stephen Warnock down the right and followed that up with a strike from the edge of the area that was comfortably saved by Guzan.
Villa did not help themselves by misplacing a number of passes on a heavy surface, but they soon began to take a stranglehold on the game without ever looking particularly impressive.
Chances soon followed as Heskey steered Stephen Ireland's inviting cross over the bar before goalkeeper Michael Poke, making just his second appearance since joining on loan from Brighton, made an excellent save from the striker as he tipped his firm header to safety.
Aaron Downes was then forced into a wild sliced clearance after Stiliyan Petrov had headed beyond the goalkeeper moments later.
With the pressure building on Rovers it was no surprise when Villa's first goal duly arrived in the 35th minute, Petrov's lovely pass allowing Albrighton in behind the defence on the right side of the area to fire through Poke's legs.
Warnock should have doubled their advantage three minutes later when Ireland's cheeky dummy left the full-back in acres of space, but he could only fire wastefully over.
Heskey was withdrawn at the interval after a recurrence of an Achilles problem, with Agbonlahor his replacement.
Rovers finally carved out their first genuine effort on goal early in the second half as Joe Anyinsah found space on the right, but substitute McGleish flashed wildly at his pull back.
That proved costly as Agbonlahor, who had looked lively since his arrival, gave Villa a two-goal cushion in the 64th minute.
The striker took advantage of a defensive slip by Cian Bolger to run in on goal and calmly guide the ball into the bottom right-hand corner for his sixth goal of the season.
Anyinsah, who was by far the home side's most threatening attacker, pushed an effort wide as Rovers continued to toil.
Villa suffered another blow when Gary Gardner replaced Collins, with the Wales international limping off with what appeared to be a calf problem.
It did little to upset Villa's comfort as Clark seemingly sealed the win with 12 minutes to go as the Republic of Ireland international cut in from the right to curl home a low left-footed shot.
But the home side got a goal back in stoppage time when Richard Dunne's woeful pass only succeeded in finding the loitering McGleish, who lofted the ball over Guzan.
Hopes of the most unlikely of comebacks were raised when Warnock capped his poor night by handling in the area seconds later, but McGleish's weak spot-kick was simple for Guzan to keep out and the chance was gone.