Grant also hit the crossbar during a dominant first-half display in Stoke Gifford with Blues showing a resolute edge to see through a vital triumph on the road.
Manager Marta Tejedor named an unchanged starting XI following Wednesday’s 1-1 draw with Tottenham Hotspur.
Hannah Hampton had an early shot across goal from Ebony Salmon to deal with but Blues started brightly and led within six minutes.
Claudia Walker’s surge down the right ended with a cutback for Whipp whose effort bent inside the top corner of the far post to hand the visitors a morale-boosting advantage.
Bristling Blues cut through regularly after that, Lucy Staniforth driving forward and shaping to shoot only to switch to a clever touch to Grant who was flagged offside just as she hammered past Sophie Baggaley.
Grant then rattled the crossbar from 15 yards with the hosts scrambling clear the follow up and when Rachel Williams dispossessed Frankie Brown in the middle, Staniforth pulled the trigger but scuffed wide.
A combination of Harriet Scott and Hampton mopped up a scruffy free-kick at the other end but Blues kept creating, Grant feeding Staniforth to again slightly miscue from distance.
Staniforth swung narrowly off target as the half came to a close with Hampton called upon to produce a routine stop from Salmon who had been steered down the left by Olivia Chance.
The hosts made a stronger start to the second half with Brown’s lob and a drive across goal from Yana Daniels bringing the best from Hampton.
However, the decisive second soon arrived when Grant took a ball from the right in her stride and squeezed a well-placed shot inside Baggaley’s left upright with 25 minutes to play.
Williams went close to a third, beating Baggaley to a loose ball but her lofted effort took a nick off Gemma Evans and dropped wide.
Former Blue Charlie Wellings unleashed a fierce dipping free-kick that Hampton had to beat away with nine minutes to go but while the Robins pressed late on there was little goalmouth action.
Bristol City: Baggaley, Matthews, Sargeant, Evans, Pattinson (Dykes, 79), Brown, Wilson, Daniels, Chance, Wellings, Salmon (Robinson, 58).
Unused subs: van der Linden, Collis, Cumings (g/k).
Bookings: Pattinson 28, Wilson 36.
Blues: Hampton, Simkin, Scott, Holloway, Jordan, Staniforth, Whipp, Arthur, Walker (Scofield, 79), Williams, Grant (Gregory, 90+3).
Unused subs: Powell, Brooks (g/k).
Goals: Whipp 6, Grant 65.
Booking: Walker 28.
Referee: Sam Allison.