Alajbegovic fired Bosnia in front in the 29th minute on Wednesday (Thursday AEST) in Seattle and Salman Al-Brake's own goal soon doubled the advantage. Assan Al-Haydos pulled one back before the break, but substitute Ermin Mahmic struck to seal a victory which puts Bosnia in a strong position to advance as one of the best third-placed teams.
Bosnia made their intentions clear as Ermedin Demirovic and Ivan Sunjic tested Mahmud Abunada inside the opening three minutes before Alajbegovic put them ahead with one of the goals of the tournament so far.
The 18-year-old, cutting in from the left, beat two defenders before hitting a rising right-footed shot into the roof of Abunada's net.
Five minutes later Bosnia led 2-0, albeit in less spectacular fashion, as Edin Dzeko's attempted cross deflected off Al-Brake to beat Abunada.
Dzeko, making his 150th appearance for Bosnia, could have killed it moments later when he beat the offside trap only to see his strike come back off a post and Qatar made the most of the let-off.
Within four minutes they were back in it as a low cross found Edmilson Junior in all sorts of space and he squared the ball for Al-Haydos to prod in. They could have been level at the break as Pedro Miguel's low strike hit a post in first-half added time.
Both sides had good chances in the second half, with Akram Afif firing into the side-netting and Alajbegovic going close to a second, but with 12 minutes left the 2022 World Cup host nation failed to clear a corner and Mahmic pounced to score from close range, sparking huge celebrations and sending Qatar home.
