The midfielder headed home a 14th-minute opener as the hosts made the most of a fast start at Kingsmeadow.
City pressed forward in search of an equaliser with Park, Hemp and Weir all going particularly close on a frustrating afternoon.
But our efforts proved elusive, as the hosts held on to preserve their slender advantage.
On a windy day at the Cherry Red Records Stadium, City almost fell afoul of the elements early on when Erin Cuthbert’s looping corner had to be cleared off the line by Georgia Stanway.
That chance set the tone for a dominant opening period from the hosts, and the pressure eventually told on 14 minutes.
Jess Carter’s inviting cross from the right was met by Guro Reiten, with her glancing header finding the far corner of Ellie Roebuck’s net despite the City stopper’s best efforts.
But Taylor’s side rallied following that setback, with Shaw’s tenacity to chase down a miscued pass from Jessie Fleming almost presenting her with a chance to grab her seventh goal in five matches moments later.
The Jamaican raced on to the loose ball after anticipating Fleming’s ball, but Ann-Katrin Berger was off her line in time to prevent her from grabbing an equaliser.
Roebuck, making her 100th City appearance, saved well with her feet from Pernille Harder’s near-post effort soon after but, having withstood the Chelsea barrage, City began to gain a foothold in the game.
Lauren Hemp demonstrated her threat from out wide when she raced away from Carter and cut the ball back into the Chelsea area, but her cross couldn’t find a City shirt just before the half hour.
It was a similar story soon after, with another inviting Greenwood set-piece presenting City with our best chance of the half when Jess Park’s goal-bound volley was expertly tipped around the post by Berger.
Chelsea were quicker out of the traps after the break, going close on three occasions to doubling their advantage inside the opening five minutes of the half.
Sam Kerr shot wide after being sent through by Harder, before the WSL’s top scorer was thwarted by a perfectly timed sliding challenge from Mace soon after.
It was Bronze’s turn for the last-ditch heroics next, blocking Reiten’s point blank effort as the midfielder seemed destined to grab her second of the game.
Hemp continued to threaten at the other end, with her ball into the far post a little too strong for Park to acrobatically steer home, while Chelsea claims of a handball inside the area from Mace fell on deaf ears.
Substitute Vicky Losada was next to try her luck, curling a vicious free-kick towards the Chelsea net with 25 to play, but Berger was always behind the effort.
As the game approached the final furlongs, City began to pile forward.
Hemp saw an effort deflected just wide, while substitute Julie Blakstad hooked Mace’s cross from the right over the bar from a promising position.
Taylor’s side were then almost prevented with a fortuitous equaliser with ten to play when Greenwood’s ball in behind bounced onto the roof of Berger’s net.
The onslaught continued when a dangerous ball in from the right from Hemp landed at the feet of Weir, but the midfielder couldn’t sort her feet out in time to pull the trigger before Chelsea recovered.
And the hosts were thankful to Millie Bright soon after when she cleared Hemp’s goal-bound volley off the line.
There was still enough time for City to go agonisingly close once again when Roebuck’s timely challenge on Lauren James sparked one final counter down the left, but Weir couldn’t find a teammate after darting in from the left.
It was a frustrating afternoon for Taylor’s side who will feel we deserved at least a point.
Player of the match: Ruby Mace
A disciplined and professional performance from the 18-year-old, who formed an impressive partnership at the heart of defence with Alex Greenwood.
Composed in possession and tenacious without, she kept the WSL's leading scorer Sam Kerr relatively quiet throughout proceedings.
How we lined up
Gareth Taylor named an unchanged side from Wednesday's 3-0 Continental Cup semi-final win over Tottenham Hotspur.
CITY XI: Roebuck, Bronze, Mace, Greenwood, Stokes, Walsh, Stanway (White 81'), Weir, Park (Blakstad 58'), Hemp, Shaw (Losada 58')
SUBS: Keating, Coombs, Beckie, Angeldahl, Raso
CHELSEA XI: Berger, Carter, Bright, Nouwen, Andersson, Cuthbert, Ingle, Reiten, Fleming (Charles 67'), Harder (Kirby 67'), Kerr (James 81')
SUBS: Musovic, Spence, Abdullina, Claypole, Thompson
What it means
Defeat sees City drop to sixth in the WSL table after Reading's 2-1 victory away to Everton.
However, we remain just five points off the Champions League places with plenty left to play for this season.
City face Manchester United at the Academy Stadium in our next WSL fixture.
The Manchester Derby gets underway at 12:30 (UK) on Sunday 13 February.
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