Burnley took the lead on 29 minutes with a piece of individual brilliance from Akinbiyi as he controlled a deflected cross from Garreth O'Connor on his chest and turned and shot in one movement, finding the corner of the net for his seventh goal of the season.
Burnley then enjoyed their best spell with O'Connor going close to extending their lead. Hull goalkeeper Boaz Myhill could only push a cross from Micah Hyde as far as O'Connor who took careful aim but placed his shot inches wide of the target.
However, the home side were indebted to keeper Jensen for preserving their lead at half time as the big Dane prevented France from getting on the scoresheet with dangerman Stuart Elliott again the provider.
Burnley manager Steve Cotterill made a tactical change at half time replacing John Spicer with striker Gifton Noel-Williams as Akinbiyi was being over-worked as their solitary striker.
The second half continued in a similar vein with Jensen stretching out a foot to stop Chris Brown from scoring after he had robbed Burnley left-back Jon Harley.
Second-half substitute Noel-Williams should have secured the points for Cotterill's side but headed wide from close range after good work from Wade Elliott.
As Hull pressed forward in search of an equaliser Stuart Elliott flicked the ball over Jensen but unfortunately just wide of the target with four minutes remaining.
Stuart Elliott was then denied a certain equaliser deep into added time with Jensen again proving his worth to the home team with another splendid save.