Ogunjimi brace bags Belgium's first points
Ogunjimi, four days short of his 23rd birthday, was on hand to knock in two rebounds after sharp shots from Racing Genk clubmate Jelle Vossen in the 52nd minute and the second after 70.
Between the goals, Belgium had a lucky escape when keeper Logan Bailly's failure to hold the ball prompted a goalmouth scramble and an apparent own goal by Toby Alderweireld, only for the referee to rule it out for handball.
Two minutes later, defender Aleksandr Kislitsyn was sent off after receiving a second yellow card, killing off Kazakhstan's fightback.
The hosts had had the best of the first 30 minutes before a sparse crowd at the Astana stadium, with Nurbol Zhumaskaliyev well placed to score but shooting over on the half-hour mark.
From that point though, Belgium, marshalled by midfielder Marouane Fellaini, gradually took over.