Quickfire goals from Lucy Graham and Ella Rutherford gave the home side an advantage at Stoke Gifford Stadium that the Reds could only halve through Courtney Sweetman-Kirk.
Rutherford had already threatened a breakthrough on 12 minutes when she neatly chested down a cross into the box, turned and struck towards goal – but Satara Murray recovered to block the drive.
Bristol did net four minutes later, though.
Graham got free to reach a delivery from the left before anybody else in the Liverpool box and thundered home an unstoppable header at the back post.
As Vicky Jepson’s Reds came to terms with their concession, City compounded the setback by claiming a second goal within five minutes.