The visitors enjoyed most of the play and territory but Blues limited the number of chances given away and went agonisingly close to pinching a late equaliser.
Head Coach Carla Ward made three changes from the midweek FA Cup semi-final exit to Everton with Georgia Brougham, Connie Scofield and Jamie-Lee Napier drafted in for Gemma Lawley, Mollie Green and Emma Kelly, all of whom were named on the bench.
Chelsea scored what proved to be the only goal nine minutes in when an inswinging free-kick from the left saw Kirby nudge her way between two defenders and her looping header dropped into the top corner of Hannah Hampton’s goal.
The visitors kept probing without much joy until a quick move started by Maren Mjelde’s surge down the right and continued via Kirby and Sophie Ingle ended with left-back Jonna Andersson slamming wide from the edge of the box.
Rebecca Holloway dug out a cutback from the left only for Lucy Whipp’s ensuing effort to get charged down.
Sarah Mayling then found Brougham on the corner of the box but the defender appeared to get caught in two minds as her cross-shot sailed over Ann-Katrin Berger’s crossbar.
Chelsea continued to see most of the ball after the break but got met with a solid blue wall when venturing towards the 18-yard box.
Kirby stung Hampton palms in the 64th minute just before a triple Chelsea change that added extra impetus to their play, Millie Bright then having a shot charged down by Rachel Corsie from a corner that was bobbling away from danger.
Beth England had the ball in the net but it was ruled out due to Sam Kerr being in an offside position and considered as in Hampton’s line of vision.
Harriet Scott pulled off a sublime goal-line clearance from England after Hampton had saved Ji So Yun’s dipping shot with 14 minutes to go.
The chance Blues had waited so patiently for came with seven minutes left, Scott driving in a cross which Claudia Walker met with a glancing header that dropped fractionally wide of the far post.
That seemed to jolt Chelsea with England and substitute Pernille Harder going close late on.
Blues: Hampton, Scott, Corsie, Brougham, Holloway, Murray, Whipp, Scofield (Green, 65), Mayling, Napier (Kelly, 68), Walker (Toussaint, 90).
Unused subs: McCarron, Grant, Lawley.
Chelsea: Berger, Mjelde, Bright, Eriksson, Andersson, Ingle (Ji, 65), Kirby (Harder, 77), Reiten (Charles, 65), Leupolz, Cuthbert (Kerr, 65), England.
Unused subs: Thorisdottir, Blundell, Carter, Fleming, Telford (g/k).
Goal: Kirby 9.
Referee: Abigail Byrne.