McQuoid, goalless in 10 games, put Eddie Howe’s side ahead when he tapped in from a yard after 53 seconds, after Liam Feeney’s deflected shot had hit a post.
Just minutes later McQuoid added his second in spectacular fashion, beating a helpless Joe Collister in the Rovers goal from 15 yards.
And with only seven minutes on the clock, Marc Pugh added a similarly stunning third with a curling effort from 25 yards.
Bournemouth continued to dominate and had chances for a fourth, but a deflected Aaron Cresswell free-kick after 27 minutes got Rovers back in it. Les Parry’s side added an unlikely second just minutes before the break when Ian Goodison rose powerfully to head home from close range.
A remarkable comeback was completed after 53 minutes when Shwan Jalal spilt another Cresswell free-kick to allow Ian Thomas-Moore to tap home, but just four minutes later McQuoid completed his hat-trick with a neat finish before Liam Feeney added a fifth with another powerful long-range effort.