The win sees Huddersfield temporarily move out of the bottom three after David Wagner's side finally picked up their first league points of 2018, thanks to goals from Mounie, club-record signing Alex Pritchard and Rajiv van La Parra at the John Smith's Stadium.
Town started the game with great endeavour and they took the lead after just seven minutes, as Mounie broke free of Simon Francis on the left, before driving into the box and slotting the ball back for Pritchard to notch his first goal for the club.
The Terriers then perhaps showed why they have conceded so often this season, when Bournemouth's Ryan Fraser crossed far too easily into the middle, leaving Junior Stanislas to wrong-foot his marker and slot coolly beyond Jonas Lossl.
Huddersfield's industry was a particular highlight in the first half, although they almost lost the lead again when a smart corner was turned into the middle for Steve Cook, who saw his glanced finish flick the far post and fly wide.
Town's energy proved contagious in the second half and they should have put any nerves to bed on the hour mark, Mooy delivering from deep to leave Tom Ince to arrive late beyond the last man, only to send his diving header over the bar from close range.
The miss was unimportant, though, as Mounie returned to bag his brace two minutes later, this time as Mooy crossed into the middle of the box, allowing the striker to drift free of his marker and clip a memorable first-time finish along the ground to make it 3-1.
Huddersfield's dream afternoon was capped off during injury time too, as Pritchard won a penalty following a clumsy tackle from Dan Gosling, allowing Van La Parra to convert from the spot.
Victory for Town, whose last league win came in December 2017, leaves Wagner's charges on 27 points and level with 14th-placed Crystal Palace.
Eddie Howe's men sit in 10th having lost their first top-flight game in eight matches.
HUDDERSFIELD TOWN (4-5-1): Lossl; Malone (Kongolo '52), Schindler, Zanka, Hadergjonaj; Hogg, Mooy (Billing '76), Van La Parra, Pritchard, Ince (Quaner '65); Mounie
BOURNEMOUTH (3-4-3): Begovic; Francis, Cook (Smith '70), Ake; Fraser, Daniels, Cook, Gosling; Ibe (King '58), Stanislas (Mousset '65), Wilson