Ex-Manchester City boss Taylor had tasted defeat in his opening three league fixtures as the Reds' head coach, but looked on course for a point until Liverpool-born Nikita Parris won a spot-kick after being brought down by goalkeeper Rafaela Borggräfe.
After a long delay, Elena Linari kept her composure and smashed the penalty high into the net via the crossbar in the 89th minute to secure a second win of the season for the Lionesses.
Tottenham Hotspur moved up to third after Cathinka Tandberg scored again to help them edge Brighton 1-0.
Summer recruit Tandberg had already scored in wins over Everton and Leicester this season and grabbed another decisive goal for Spurs when she finished off a smart cross by Jessica Naz after 26 minutes.
It proved enough to ensure Spurs continued their fine start under Martin Ho, with the north London club onto 12 points and up to four wins -- only one short of their overall tally in the 2024-25 campaign.
Second-half goals by Kirsty Hanson and Lynn Wilms helped Aston Villa register a first league victory of the season with a 2-0 triumph at West Ham.
Hanson broke the deadlock on the hour mark with a superb strike after a short corner before provider Wilms produced an equally excellent free-kick seven minutes later to wrap up the scoring.
The result marked a fifth consecutive league defeat for West Ham and kept them bottom of the table.
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Noémie Mouchon's 81st-minute strike rescued a late point for Leicester at home to Everton.
Ornella Vignola finished off a quick counter-attack for the Toffees after 51 minutes at the King Power Stadium to open the scoring.
The visitors pushed for a second but were hit late on when Mouchon scored to ensure the spoils were shared.
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