The Potters' record signing, who arrived in January, powered home to give the visitors a half-time lead.
Ibrahim Afellay struck a sweet second for Stoke, and substitute Joselu made it 3-0 two minutes later.
Matt Ritchie responded immediately for the Cherries, but his left-footed strike proved to be a mere consolation.
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It was Stoke's first away win in 2016 and only their second league victory in seven attempts.
Mark Hughes's side move up to ninth in the table, five points off the European places, while Bournemouth remain 15th and four points above the relegation zone.
Relive Stoke's win over Bournemouth
Follow reaction to Saturday's games
Stoke had been on the end of three consecutive 3-0 defeats in the league, and had failed to get on the score sheet for a month.
But they looked clinical from the outset at the Vitality Stadium, even without Marko Arnautovic and Bojan in the side.
Frenchman Imbula, who signed from Marseille for £18.3m on transfer deadline day, opened his Stoke account in only his second appearance by racing onto the ball and thrashing home.
A superb counter-attack led to their second, as the impressive Jon Walters held the ball up before feeding to Afellay, who swept home his first ever Premier League goal.
And only moments after coming on for the injured Glen Johnson, Joselu grabbed a third with a rising header after a pinpoint ball from Xherdan Shaqiri.
Bournemouth enjoyed 62% of the possession in the first-half but failed to really test Stoke goalkeeper Jack Butland - bar a looping effort from Charlie Daniels.
Benik Afobe had scored three in five games since joining from Wolves in January but looked out of his depth at times, and only managed two touches in the box during the first 45 minutes.
Bournemouth boss Eddie Howe brought on Scotland international Ritchie for the second half, along with Josh King, and the wide pair made a huge difference going forward.
Ritchie scored with a clinical effort from the edge of the box, and he almost grabbed a second from King's ball in, as Stoke struggled to cope without injured defenders Johnson and Marc Muniesa.
But eventually the Cherries faded and they were unable to pull off an unlikely comeback, leaving them with back-to-back league defeats.
Bournemouth manager Eddie Howe: "They were quite compact and deep and I thought they were very respectful of how we played and the first moments of the game I was expecting us to have a really good afternoon but the goal seemed to change the game.
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"I think the quality of their goals was really high and their first goal was a great strike, their third goal was very well worked but their second goal was the one I think we are really disappointed with."
Stoke boss Mark Hughes: "It was a really pleasing away performance, we came here with a good game plan.
"We conceded a bit of possession, more than we usually do but we felt we could get good opportunities on the break. With the personnel that we have got we could cause Bournemouth a problem."
Bournemouth welcome Everton in the FA Cup fifth round before travelling to Watford in the league the following week.
Stoke have a week off before hosting bottom club Aston Villa on 27 February.
Match ends, Bournemouth 1, Stoke City 3.
Second Half ends, Bournemouth 1, Stoke City 3.
Juan Iturbe (Bournemouth) wins a free kick on the right wing.
Foul by Erik Pieters (Stoke City).
Attempt missed. Xherdan Shaqiri (Stoke City) left footed shot from more than 35 yards misses to the left.
Giannelli Imbula (Stoke City) is shown the yellow card for a bad foul.
Harry Arter (Bournemouth) wins a free kick in the defensive half.
Foul by Giannelli Imbula (Stoke City).
Attempt saved. Jonathan Walters (Stoke City) right footed shot from the centre of the box is saved in the bottom right corner. Assisted by Stephen Ireland.
Harry Arter (Bournemouth) is shown the yellow card for a bad foul.
Foul by Harry Arter (Bournemouth).
Xherdan Shaqiri (Stoke City) wins a free kick in the defensive half.
Adam Smith (Bournemouth) wins a free kick in the attacking half.
Foul by Xherdan Shaqiri (Stoke City).
Corner,  Bournemouth. Conceded by Dionatan Teixeira.
Attempt blocked. Steve Cook (Bournemouth) right footed shot from outside the box is blocked.
Adam Smith (Bournemouth) wins a free kick on the right wing.
Foul by Erik Pieters (Stoke City).
Benik Afobe (Bournemouth) wins a free kick in the attacking half.
Foul by Glenn Whelan (Stoke City).
Substitution, Stoke City. Stephen Ireland replaces Ibrahim Afellay.
Simon Francis (Bournemouth) is shown the yellow card for a bad foul.
Foul by Simon Francis (Bournemouth).
Jonathan Walters (Stoke City) wins a free kick on the left wing.
Matt Ritchie (Bournemouth) wins a free kick in the defensive half.
Foul by Giannelli Imbula (Stoke City).
Corner,  Bournemouth. Conceded by Glenn Whelan.
Charlie Daniels (Bournemouth) wins a free kick on the left wing.
Foul by Xherdan Shaqiri (Stoke City).
Adam Smith (Bournemouth) wins a free kick in the defensive half.
Foul by Joselu (Stoke City).
Attempt missed. Harry Arter (Bournemouth) left footed shot from outside the box is high and wide to the left following a corner.
Corner,  Bournemouth. Conceded by Philipp Wollscheid.
Matt Ritchie (Bournemouth) wins a free kick on the right wing.
Foul by Erik Pieters (Stoke City).
Foul by Simon Francis (Bournemouth).
Erik Pieters (Stoke City) wins a free kick in the defensive half.
Substitution, Bournemouth. Juan Iturbe replaces Junior Stanislas.
Andrew Surman (Bournemouth) is shown the yellow card for a bad foul.
Foul by Andrew Surman (Bournemouth).
Giannelli Imbula scored his first goal for Stoke as they ended their three-match losing streak in the Premier League against Bournemouth.