Bournemouth led at the interval thanks to a Jenna Markham goal, but Wimbledon hit back to equalise early on in the second period.
But that sparked the Cherries into life, and Quirk scored a hattrick and Kenni Thompson scored a penalty to earn a resounding win for the Dorset side.
The gap to Argyle in the league remains at six points, with two games in hand in favour of Bournemouth, as the Devon side defeated Oxford 5-0 on Sunday.
The hosts at the Ringwood Community Hub made a good start as they went close in a goalmouth scramble, but keeper Rosie Kite kept it out.
The Wombles went close themselves from a corner after eight minutes, before Kite was at her best to deny Georgia Wilson with a half volley.
Jenna Markham. (Image: Newsquest)
But Bournemouth took the lead after 21 minutes as Markham directed a header into the back of the net from the top of the box.
After the Cherries saw a goal disallowed for offside, the visitors went close but Katie Scadding was first to the ball ahead of Kaylee Buckingham.
Emily Donovan shot wide for Wimbledon as Bournemouth entered the interval in front.
Bleazard introduced Sophia Leonard and Freya Meadows Tuson for Kelci Bowers and Charlotte Harris at the break, and it was the worst possible start to the half for the Cherries.
Wimbledon drew themselves level just two minutes after the resumption, as Kaylee Buckingham was first to the loose ball when it dropped in the box after a corner, and she finished past Scadding.
But Bournemouth sought to get themselves back in front and they did just that shortly after the hour mark.
Quirk was the scorer as met a Markham cross from the left with her head to guide it backwards into the goal and make it 2-1 with 64 minutes played.
And it wasn’t long before the advantage was extended to two goals, as Thompson cooly converted from the spot four minutes after the second after the Cherries were awarded a penalty.
Sophie Quirk. (Image: Newsquest)
Phoebe Williams replaced Molly Barron-Clark before Quirk snatched her second of the afternoon as she crossed from the right after a good run and her effort found the back of the net.
After that, Gemma McGuinness and Zoe Barrett replaced Markham and Thompson, before Quirk completed her hattrick deep into stoppage time.
The Cherries top scorer stepped up to take a free kick from just outside the box and she fired into the back of the net to score her third of the afternoon and complete an impressive win for Bournemouth.
The victory moves them onto 43 points from 19 games, while Plymouth ahead are on 49 from 21.
Up next, the Cherries host Real Bedford at the Vitality Stadium next weekend.