Kerr secured a hat trick and an assist while Kirby scored two and set up the other four including Guro Reiten's 55th-minute strike.
"It was a massive team performance. We really deserved it. It was important to score early," Kerr said after the game.
"Fran Kirby has been unbelievable and she has been putting it on a plate for whoever has been playing up front all season.
"She is quality on the ball. She is a special talent.
"We are not stopping here, but we will enjoy it tonight. It's about winning all we can."
Bristol City came into the clash as massive underdogs, scoring just once in their last four games against Chelsea while their rivals put 24 goals past them including a 9-0 win in September.
Kerr needed just 90 seconds to open the scoring for Chelsea. In what has become a typical sight this season, Kirby scampered down the wing and crossed the ball into Kerr who coolly slotted it past Sophie Baggaley and celebrated with her trademark back flip.
A misplaced kick out from Baggaley on 29 minutes handed it straight to Kirby who chipped the ball over her head and into the empty net.
Kerr repaid Kirby for her earlier service by putting her through for her second and Chelsea's fourth on 35 minutes. Baggaley did get a hand to the ball but it spun away from her and into the net.
The switch up didn't affect the dynamic duo of Kerr and Kirby, however, and the Australia forward got her hat trick four minutes into the second half thanks to a set up from Kirby.
Another Kerr-Kirby link up on 55 minutes saw the latter play Reiten in on goal who placed her shot past Baggaley.
There were few bright sparks for Bristol City, who are facing a relegation battle in the Women's Super League. Young talent Ebony Salmon had a great opportunity to pull one back for Bristol City just before the hour mark but Ann-Katrin Berger pulled off a good save to deny her.
The game ended on an unfortunate note when Maren Mjelde went down under a heavy tackle and had to be stretchered off after a long period of treatment on the pitch. BT Sport reported that she was taken straight to hospital from the ground.