The Clarets went 2-0 up against the run of play before having goalkeeper Brian Jensen sent off for deliberate handball before the break.
Steve Howard's 95th career goal made it 2-1, but Akinbiyi stroked home a penalty before Warren Feeney's sixth of the season reduced the deficit only for the visitors to hang on.
The game's statistics told the story as Luton had ten shots on target, 11 shots off target and 13 corners compared to Burnley's four attempts on goal, two attempts off target and three flag-kicks.
The result was Hatters' first home league defeat since the 3-1 reverse to Barnsley on Good Friday in March.
The Hatters started without first-team regulars Kevin Nicholls, Chris Coyne, Steve Robinson and Sol Davis, meaning Howard wore the captain's armband on his 200th Luton appearance.
Despite the enforced changes it was Town who started the better with Dean Morgan, Warren Feeney, Carlos Edwards and Howard all firing efforts straight at visiting goalkeeper Jensen within the opening 13 minutes.
But it was the Clarets who took the lead amid a touch of controversy in the 15th minute when John Spicer appeared to handle the ball before slipping in Akinbiyi, who raced clear before cleverly firing the ball past former Burnley stopper Marlon Beresford.
The hosts pressed again with Edwards hitting another effort straight at Jensen and Howard forcing a superb save from the Clarets custodian from a 20-yard free-kick before Burnley shocked the Hatters with a second in the 31st minute.
There seemed little danger as Edwards dribbled the ball out of defence, but the Trinidad & Tobago international gifted the ball to Spicer, whose first-time pass again set Akinbiyi free and the former Stoke man finished in style as he slammed into the top corner from 16 yards.
Luton looked shell-shocked, but were handed an unexpected lifeline six minutes before half-time when Jensen inexplicably handled a Kevin Foley pass outside his area and was instantly sent off.
With no goalkeeper on the bench, Burnley were forced to put Spicer in goal and four minutes later the stand-in keeper was beaten as Howard nodded in a Morgan cross for his seventh of the season.
The second period should have seen Luton come out all guns blazing, but instead it was the visitors who frustrated the hosts and unbelievably stretched their lead in the 55th minute.
Youngster Leon Barnett horribly mistimed a tackle on Akinbiyi to concede a penalty and was booked for his trouble and the Burnley striker stepped up to blast home from the spot for his hat-trick.
Luton came again though and six minutes later they were back in it as Howard headed down another Morgan corner and Feeney pounced to sweep home the rebound from six yards for his sixth of the season.
The hosts failed to build on the goal, however, with the closest they came to testing Spicer again being a 25-yard Edwards shot that screamed past the post in the 76th minute.