The visitors took the lead on 18 minutes when Kevin Phillips curled a shot into the corner of the goal.
But Burnley drew level when Michael Duff rose highest to head home from a corner just after half-time.
And substitute Akinbiyi's strong running forced Neil Clement into a handball which allowed Andy Gray to score the winner from the penalty spot.
West Brom manager Tony Mowbray:
"It was a typical first game of the season - kick and rush and very hurly-burly.
"You get that on the opening day of the season but there are no excuses about teams settling down.
"In the first half there were some good signs there but a lack of concentration for the goals cost us, and that was very similar to a number of games last season."
Burnley: Kiraly, Duff, Thomas, Caldwell (Lafferty 31), Jordan, Elliott, Gudjonsson, McCann, Harley, Blake (Akinbiyi 73), Gray.
Subs Not Used: Jensen, James O'Connor, Spicer.
Goals: Duff 47, Gray 79 pen.
West Brom: Kiely, Hoefkens, Barnett, Clement, Tininho, Gera, Chaplow (Morrison 70), Greening, Phillips, Beattie (Ellington 70), Filipe Teixeira (Pele 64).
Subs Not Used: Steele, Albrechtsen.
Goals: Phillips 18.
Ref: E Ilderton (Tyne & Wear).