Callum Wilson was denied what appeared a clear penalty in the early stages, but on-loan defender Ake gave Bournemouth a deserved lead by marking his first start for the club with a goal on 26 minutes.
After Ryan Shawcross had a header cleared off the line before the break, Bojan's penalty clattered off the woodwork after he had been fouled by Steve Cook just a few minutes into the second-half.
The home side saw chances come and go throughout the rest of the second period, but their inability to make the most of them meant Bournemouth claimed their fourth win of the season and kept a first clean sheet in 10 away league games in the process.
The win lifts Bournemouth to ninth in the table, while Stoke - whose six-game unbeaten run was ended here - drop down to 13th.
A couple of changes for both sides
Mark Hughes made three changes from the side that drew away at West Ham United before the international break, Xherdan Shaqiri replacing Jonathan Walters after recovering from a hamstring injury.
Bojan came in for just his fourth start of the campaign, occupying the No.10 role which Joe Allen has thrived in this term, while the Welshman dropped into a deeper midfield role with Glenn Whelan out.
Ramadan Sobhi dropped to the bench to accommodate the returning Marko Arnautovic, who served a one-match ban last time out.
The visitors handed goalkeeper Adam Federici and Ake - on loan at the Vitality Stadium from Chelsea - their first Premier League starts of the season, with Adam Smith suspended and Artur Boruc failing a late fitness test.
Eddie Howe - taking charge of his 200th game as Cherries boss - also handed a start to Callum Wilson, who missed their defeat to Sunderland a fortnight ago. Jordon Ibe was unavailable due to illness, with Josh King switching to a wide role.
Bournemouth fans were vocal in their discontent of the early decisions of referee Roger East and understandably so. Wilson was contentiously flagged offside inside a minute as he raced clear through on goal.
And on 11 minutes East should have pointed to the spot, as Wilson - once again through one-on-one after a fine counter-attack - went down under the challenge of Shawcross.
Though the Stoke skipper got none of the ball, the match official waved away the away side's shouts for a penalty.
But as much as they had to complain of the referee's decision-making, Bournemouth also had their own inability to capitalise on their chances to rue.
Jack Wilshere shrugged off the attention of Allen and found King running free into space the box, but rather than shoot first time, the striker took a touch which allowed Lee Grant to get down low and push his shot past the post.
Cherries open the scoring through Ake
The Cherries continued to look the more likely to open the scoring, with Stoke's final ball repeatedly letting them down up the other end, and it was no surprise when they broke the deadlock on 26 minutes.
Steve Cook was awarded a free-kick after a tussle with Erik Pieters on the by-line down the right-hand side - much to the despair of the Stoke supporters, who felt Cook had gone down too easily.
From the following set-piece, no defender tracked the run of Ake, who had no trouble in heading beyond Grant from just six yards out to make it 1-0.
Stoke's struggles in the final third saw them limited to few real first-half chances, Shaqiri curling a free-kick over the bar after Allen had been fouled on the edge of the area.
But they enjoyed a more threatening spell in the final few minutes before the interval, only the head of Cook denying them an equaliser as he cleared Shawcross' header at the second attempt off the line at a corner.
Arnautovic was later guilty of rushing his shot when in space inside the area, Federici only needing to make a simple save to deny the Austrian winger, as Stoke were left frustrated going into half-time.
Stoke spurn chance to draw level as Bojan misses spot-kick
And those frustrations continued into the second-half. Wilfried Bony's header from a Charlie Adam cross was pushed out to Bojan, who was brought down by Cook clumsily treading on the Spaniard's ankle from behind, and East rightly pointed towards the spot.
But following lengthy build-up, in which Federici was booked for time-wasting, Bojan cannoned his attempt off the crossbar as the 'keeper dived to his left, before the rebound was scrambled clear by Bournemouth.
Stoke's misery was compounded when Phil Barsdley went off injured before the hour, and the hosts were fortunate that Wilson failed to make more of Wilshere's cross soon after.
Frequent pauses in play disrupted the tempo of the game, barring King's shot into the side-netting from a tight angle, frustrating the home crowd even further.
On 67 minutes, Hughes turned to Peter Crouch off the bench for the third consecutive game as he replaced Bony, looking to provide more of a focal point up top.
And the Welsh boss also brought on Walters for Bojan as he desperately seeked to turn the tide in Stoke's favour, but neither substitution really worked as they threw everything at their South Coast oppositon.
Wasteful Potters made to rue missed opportunities
Tempers frayed late on as the crowd just as desperately willed Stoke on, but their poor passing and wastefulness in front of goal let them down time and time again.
Shaqiri spurned a gilt-edged chance on 81 minutes after Arnautovic broke free down the left and pulled the ball back to the Swiss international, but he couldn't keep his shot down from 12-yards.
Chances came and went for Stoke in the final stages, Walters' tame header the most notable, but a tired Bournemouth side just about clung on in the final stages to earn the club their first-ever top-flight victory in the month of November.