But the Premiership side were made to fight all the way by a battling performance from League One outfit Scunthorpe.
Manager Martin O'Neill named a full strength side for the trip to Glanford Park, but had an anxious moment in the opening minute when goalkeeper Thomas Sorensen was forced to save bravely at the feet of striker Billy Sharp.
Andy Keogh also tested Sorensen with another good effort as Scunthorpe started brightly.
But after an uncomfortable opening half hour it was Villa who began to take charge.
Luke Moore was denied by a good save from goalkeeper Joe Murphy before Angel rattled a shot against the bar in the 33rd minute.
Murphy was tested again by another fierce drive from Angel, before Villa took the lead from a penalty after Richard Hinds had pulled down Moore.
Angel's spot-kick was kept out by the legs of goalkeeper Murphy, but the Villa striker followed up to net the rebound.
Scunthorpe made a good start to the second half and Villa had an escape when Sorensen fumbled a 20-yard drive from Keogh, before recovering to block away a follow-up effort from Sharp.
Villa then pushed forward and, after Gabriel Agbonlahor had wasted two good chances, it was Angel who made it 2-0 in the 63rd minute when he rose unmarked at the far post to head home a right-wing cross from Aaron Hughes.
That looked to be Villa safely through, but Scunthorpe had other ideas and came roaring back in the 74th minute to reduce the arrears.
Ian Morris swung in a corner from the right and in a crowded box it was Sharp who headed past Sorensen from close range for his sixth goal of the season.
Sharp also tested Sorensen again with a long-range effort, but it was Villa who eventually won through to round three of a competition that has twice been won by manager O'Neill.