The game finished goalless in normal time as both sides cancelled each other out, before the Foxes triumphed in the shootout to earn an extra bonus point and top the group ahead of their Barclays FA Women's Super League opponents.
Following Sunday's 3-0 league victory over London Bees, Jonathan Morgan opted to name an unchanged team for the midweek trip to Solihull.
The opening exchanges were devoid of any clear-cut chances, but the best moment for the Foxes came when the ball fell for Charlie Devlin, who tried to lob Blues 'keeper Hannah Hampton, only to see it rise over the bar.
And when Natasha Flint found space for a shot from range, her strike also narrowly missed the target. At the other end, Kirstie Levell was called into action to deny Georgia Brougham with a smart stop.
More space started to open up as the half wore on and a well-worked City move almost brought the opener. Esmee De Graaf moved down the left and picked out Paige Bailey-Gayle in the middle, who laid off Devlin to shoot low, but it was straight at Hampton.
On 37 minutes, the hosts were handed a golden opportunity to break the deadlock when Lucy Whipp's shot was adjudged to have hit the arm of Remi Allen in the area. Mollie Green stepped up and Levell pulled off a brilliant diving save to keep the scores level.
A flurry of chances followed for the visitors as Flint sliced her shot wide after appearing to be pushed by Hampton in the box, before Bailey-Gayle went close and Lachante Paul forced Birmingham's shot-stopper into a save.
The first second-half opportunity fell the way of Birmingham's in-form striker Claudia Walker, who fired just over after turning on the edge of the area, while Sam Tierney had to make a superb block to deny Emma Kelly, whose shot on the turn looked to be goal bound.
The Foxes also went close, first through Paul, who drove forward before striking at Hampton and then via Bailey-Gayle, who was unable to add to her recent tally, firing high and wide once getting through on goal.
There was still no way through for City when Allen met De Graaf's centre and found Hampton in the way again, before Ashleigh Plumptre twice headed Sophie Barker's pinpoint corners on target, but there were blocks in the way.
Bailey-Gayle then had an effort cleared off the line as the visitors came within inches of an opener. Meanwhile, Levell had to be alert to keep out Christie Murray, but it was City that were doing the majority of the pressing.
The Leicester No.17 was in the thick of the action once again, but when she got in behind the Birmingham defence, she dragged her shot wide, before the hosts got bodies in the way of Millie Farrow's attempt, in what was City's final chance, sending the clash to penalties.
Once Connie Scofield hit the bar, 19-year-old defender Aimee Everett was calm and composed to step up and find the target, sending City through, much to the delight of the Foxes players and staff.
City's stopper made some smart saves throughout the contest, including a first half spot-kick save.
It's hard to look past the first-half penalty save from Kirstie Levell, which prevented what could have been a winner for the hosts.
She made an excellent diving stop to keep out Mollie Green and although the Foxes pushed for a winner in the second half, it was that spot-kick save that gave the team a chance to go and win the shootout.
Sam Tierney helped City to keep a third clean sheet on the bounce in all competitions.
There were plenty of bright sparks within the 11, but centre-back Sam Tierney produced another assured display alongside Ashleigh Plumptre in the middle of defence to keep one of the WSL's in-form strikers, Claudia Walker, at bay.
And after being substituted late on, it was Tierney's replacement, Aimee Everett, who scored the winning spot-kick in the shootout. Tierney and the rest of the defence will also be happy with a second consecutive clean sheet, as they continue to build momentum in both league and cup.
Where does it leave us?
Foxes defender Aimee Everett scored the winning spot-kick after coming off the bench in stoppage time, leading to jubilant celebrations.
Leicester have qualified for the last eight of the competition for the first time in the Club's history and now await their quarter-final opponents.
The Foxes have finished top of the three-team group on five points from two games, beating Blackburn Rovers and then taking two points from the draw with Birmingham after winning the penalty shootout.
Kirstie Levell celebrates with her team-mates after playing a big role in the victory.
City head to The Oakwood on Sunday for a FA Women’s Championship clash with Charlton Athletic (2pm kick-off GMT).
City: Levell; Smith, Tierney (Everett 90), Plumptre, De Graaf (Blanchard 90); Barker; Allen, Devlin; Bailey-Gayle, Flint (Farrow 65), Paul.
Subs not used: Riglar, Cataldo, Howard, Thomas, Fergusson, Wood.
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