Sherwood had a watching brief but after a goalless opening 45 minutes, his half-time team-talk seemed to galvanise Villa and goals from Leandro Bacuna and Scott Sinclair settled a scrappy game
Andrej Kramaric netted an injury-time consolation for the Foxes but they could not force a replay.
First-team coach Scott Marshall took charge of Villa after ex-Tottenham boss Sherwood was appointed as Paul Lambert's successor on Saturday.
Marcin Wasilewski hit the post for Leicester and Shay Given denied Matty James while Christian Benteke and Andi Weimann had goals ruled out for offside for the hosts.
But Villa reached the last eight for the first time since 2010 despite their late scare.
Benteke, Bacuna, Ron Vlaar, Tom Cleverley and Given all returned for Villa while Leicester replaced the cup-tied Robert Huth with Wasilewski.
Foxes boss Nigel Pearson was also in the stands and would have been worried by the Foxes' slow start.
Eager to impress, Villa opened well and Weimann blazed over from 16 yards but their attacking threat waned, showing Sherwood the immediate problem he needs to fix.
While there was plenty of endeavour the hosts lacked quality with Benteke isolated and unable to seriously trouble the Foxes defence.
It gave Leicester encouragement and Wasilewski glanced James' corner across goal and onto the face of the far post after 12 minutes.
Slowly, the Foxes began to assert themselves and Kramaric went down under pressure from Vlaar before Wes Morgan headed over.
Leicester were on top and Villa needed Given to roll back the years after 31 minutes to deny James.
Jeff Schlupp's pass was dummied by Esteban Cambiasso for James to arrow a shot towards the top corner, only for Given to produce a wonderful fingertip save.
Villa did not need another warning and thought they were ahead a minute before the break when Benteke swept into the corner, but his strike was ruled out for offside.
Seemingly buoyed by Sherwood's half-time words, the hosts began the second half well and Fabian Delph's mis-hit header flew over before Mark Schwarzer saved at Benteke's feet after Vlaar's header caught out the Leicester defence
The Foxes looked ragged and Benteke headed wide before Villa finally made their dominance count after 68 minutes.
Vlaar slipped a cute ball through to Bacuna and the midfielder cut inside to unleash a curling effort into the bottom corner from 18 yards.
Two minutes later Weimann had the ball in the net, only to be pulled up for offside, and Benteke fired over after outpacing Matthew Upson.
A dour game had come to life and Leicester almost hit back when Kramaric failed to force the ball in after Given had spilled a cross.
Vlaar then denied Riyad Mahrez with a last-ditch tackle and Kramaric headed over.
But Villa made it 2-0 in the 89th minute when substitute Sinclair wriggled free
His weak shot was straight at Schwarzer but the keeper spilled it into his own net.
Leicester immediately hit back in stoppage time when Kramaric headed in Schlupp's cross but Villa held on.
Source : PA
Source: PA