In her first start since October, Bunny Shaw starred, bagging two goals before half time as Gareth Taylor's side came from behind after the visitors had broken the deadlock through an Alex Greenwood own-goal.
A dominant performance and a third goal, courtesy of Vicky Losada, helped City overcome the early set-back to extend our winning run to six, ending Bristol's nine-game unbeaten streak in the process.
City came out strong with good early possession which led to two quick chances - both from Janine Beckie crosses. The first deflected through the box but Shaw couldn’t get a good connection and Bristol 'keeper Sophie Whitehouse collected; the second landed again out of Shaw’s grasp and Whitehouse was there to clean up.
Against the run of play, Bristol made a break for goal and though Greenwood got in the way to stop the cross reaching one of the strikers, an unfortunate deflection forced the ball past Khiara Keating into the net.
However, City rallied and began to mount the pressure.
Houghton’s header was forced over the bar for another corner before our domination paid off, with Shaw calmly slotting into the bottom left corner for the equaliser.
Beckie then curled a shot just wide of the far post before Angeldahl dragged an effort just wide from a corner.
But Taylor's side deservedly got our noses in front when Shaw latched on to Bronze's pass and finish smartly for her second of the game.
It was almost three moments later as Hemp rifled an effort off the underside of the bar, but City had the lead going into half time.
After the break, we picked up where we left off with strong pressure in and around the Bristol goal, with Shaw denied a hat-trick thanks to a smart stop from Whitehouse.
City continued to cause headaches for Bristol when a corner taken by Losada was lofted into the box and found Greenwood but Whitehouse managed to collect the ball on the goal line to keep her team in the game.
City then had another three good chances in quick succession as we looked more and more threatening in our search for a third.
First, a powerful strike from Lucy Bronze was deflected away by Whitehouse before Shaw's effort fell to Losada who put it just wide of the goal
The constant pressure from City was finally rewarded with a goal as Losada capitalised on a communication error from the Bristol back line and volleyed home from close range.
A mistake by Whitehouse allowed for Jess Park to get a good shot away but the keeper recovered in time to make the stop, as the onslaught continued.
Beckie and Angeldahl both fired off-target as City went in search of another, but three would prove enough to progress on a pleasing night.
The draw for the semi-finals will be held on Thursday at 3.30pm on talkSport 2.
How we Lined up
Gareth Taylor made four changes to the team that beat Aston Villa 3-0 on Sunday with Laura Coombs, Filippa Angeldahal, Janine Beckie and Bunny Shaw coming in to replace Demi Stokes, Ellen White, Keira Walsh and Jess Park.
CITY XI | Keating, Stanway, Greenwood, Houghton (C) (Mace 46'), Bronze (Stokes 75'), Coombs (Weir 62'), Angeldahl, Losada, Hemp (Park 75'), Beckie, Shaw (White 62')
SUBS | Pilling, Taieb, Scott
BRISTOL XI | Whitehouse, Powell (Allen 70'), Cataldo, Aspin, Palmer (C), Harrison (Woolley 80'), Johnson (Morgan 46'), C Bull, J Bull (Davies 61'), Beever-Jones (Cook 61'), Murray
SUBS | Bentley, Jhamat, Allen, Kuyken, Collis
Player of the match: Bunny Shaw
In her first start since October, she produced two pieces of brilliance to help City overcome Bristol.
Bunny really showed what she's capable of, providing the spark the team needed with the two goals and a number of chances in the game.
City welcome Arsenal us a crunch FA WSL clash for our first home league game of 2022.
Don't miss your chance to cheer on Gareth Taylor's side on Sunday 23 January. Kick-off 18:45 at the Academy Stadium.
Our meeting with Bristol will be our 17th in all competitions. We have won 13 of those encounters and have played them three times in the Conti Cup, triumphing on all three occasions.
Bunny's two goals in this game means she has now scored eight in 14 games.
Losada's goal tonight was her fourth goal in four games.
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