Lisa Naalsund opened the scoring after just nine minutes, as she weaved between the Robins' defence before placing a low effort that beat goalkeeper Shae Yanez.
Our no.16 grabbed a second with just minutes remaining to complete a fine brace, hooking the ball home from Katie Zelem's set-piece delivery.
United finished the game with a player advantage, after Bristol City's Jamie-Lee Napier was shown a red card in the closing stages, having picked up a second booking for a foul on Jayde Riviere.
FIRST HALF
After just nine minutes, United took the lead thanks to a well-worked goal from Naalsund.