First-half goals from Sam Vokes, Jo Kuffour and Max-Alain Gradel put the Cherries - deducted ten points for entering administration on Friday - in the driving seat.
Lewis Emanuel pulled one back for the hosts in the 50th minute, but Kuffour put the seal on the win seven minutes from time with his second of the afternoon.
Luton, already missing their only recognised centre-half Chris Perry, suffered a blow ten minutes in when Keith Keane was forced to limp out of the action to be replaced by 17-year-old George Beavan on his full debut.
Things got worse for Luton as the visitors struck twice in the space of a minute just moments later.
The first goal came in the 13th minute when a tackle between Lee Bradbury and Sam Parkin in the centre of the park saw the ball spin behind the static home defence for Vokes to run in and round Dean Brill to score.
Sixty seconds later it was 2-0 as a long punt upfield from Danny Hollands produced some comical Town defending.
Alan Goodall fell over, missing his attempted header, and Kuffour nicked the ball over Emanuel before dinking past the advancing Brill from 10 yards.
The Hatters were at sixes and sevens and Bournemouth could have scored more as Vokes was denied by Brill in the 17th minute and ex-Luton man Russell Perrett nodded just wide at the far post in the 22nd minute.
Luton briefly rallied but it was 3-0 at the break as Gradel hit a 30-yard beauty into the top corner a minute before the interval after Vokes' nod down.
The hosts made a change at half-time, bringing on Calvin Andrew for Darren Currie and five minutes later they pulled a goal back when Emanuel's speculative 35-yard low shot sneaked in under Gareth Stewart at the goalkeeper's near post.
Luton plugged away with little success and instead Kuffour almost made it 4-1 when his volley flew back off the crossbar in the 81st minute.
Kuffour did find the net seven minutes from time when he converted from close range after his first shot had come back off the post.