Linnett scored in the 36th, 49th and 57th minutes - with the second a stunning effort from 35 yards.
Kerys Harrop had opened the scoring and Charlie Wellings slid home to make it 3-0 at the break.
Beth Mead pulled one back from 25 yards out before Freda Ayisi and Coral Haines added to Birmingham's tally.
Kirsty Linnett scored a hat-trick as Birmingham City capped off their Women's Super League One season with a thumping win at Sunderland.