Dwight Gayle gave the Baggies an early lead on Saturday afternoon but was dismissed for two yellow cards in the closing stages, and two goals during a four-minute period in the second half from Conor Hourihane and Tammy Abraham sealed the win for the home side.
Villa started brightly with Glenn Whelan sending a header over the crossbar from a well-worked corner, but West Brom did not take long to pose a threat at the other end.
Abraham volleyed over from 12 yards for Villa as both teams pressed for an early goal, but it was West Brom who made the breakthrough after a mistake from Whelan.
The midfielder miscontrolled a pass around 35 yards from goal, and it allow Gayle to run towards the area before lashing a low 18-yard strike into the bottom corner.
Villa continued to dominate possession after going behind, but the home side's best opening came when Jack Grealish drilled a shot over the crossbar after a short corner.
West Brom remained a threat on the counter-attack, but the Baggies were prepared to manage the game efficiently with the onus on Villa to make something happen going forward.
Villa went onto the front foot after the restart as Abraham got himself in the thick of the action, with a shot at the near post being saved by Sam Johnstone and a glancing header being directed just wide of the far post.
Anwar El Ghazi also went close with a speculative effort from 30 yards out, but West Brom did not give up on doubling their advantage with Kyle Bartley heading marginally wide of the far post from a corner.
Time was beginning to run out for Villa but an equaliser arrived through the left boot of substitute Hourihane, who sent a first-time strike from 20 yards into the far corner after being set up by Grealish.
Villa's star man also played a role in his team's second goal. The playmaker was tripped by Kieran Gibbs inside the area, and Abraham stepped up to slot the ball to Johnstone's left.
Things went from bad to worse for West Brom with two minutes remaining, as Gayle was given a second yellow card for diving into Steer when attempting to meet a cross, although there was an argument that the collision was nothing more than accidental.
West Brom survived seven minutes of added-on time, but James Shan's team will be without their top goalscorer for Tuesday's second leg at The Hawthorns.
ASTON VILLA (4-3-3): Steer, Elmohamady, Tuanzebe, Mings, Taylor; Whelan (Hourihane 67'), McGinn, Grealish; Adomah (Green 67'), Abraham, El Ghazi (Kodjia 91')
WEST BROMWICH ALBION (3-4-1-2): Johnstone; Dawson (Mears 46'), Bartley, Hegazi; Holgate, Brunt, Johansen (Morrison 85'), Gibbs; Phillips (Murphy 66'); Rodriguez, Gayle