The attacker's fortunate finish in the 24th minute settled a rather unspectacular contest, with the hosts left to rue several unsuccessful sequences of play in the final third as their 14-game European winning run at home came to an end.
Mikel Arteta named a stronger-than-expected lineup for Thursday's game, but there was little in the way of early clear-cut chances as Arsenal became accustomed to the artificial surface.
The Aspmyra Stadion faithful were unsurprisingly in full voice as their side frustrated Arsenal in the opening exchanges, but Arteta's side soon got their desired goal in unorthodox circumstances.
A neat link-up between Albert Sambi Lokonga and Saka ended with the latter firing a shot towards goal from close range, and his effort deflected off of a Bodo/Glimt defender onto his face, and the subsequent touch directed the ball into the back of the net.
The 21-year-old knew very little about it, but Arsenal were in the ascendancy with 24 minutes gone, although Kjetil Knutsen's side missed a gilt-edged chance to level the match straight away.
A missed header from Rob Holding opened up space for Amahl Pellegrino to run through one-on-one against Matt Turner, but the 32-year-old miscued his curling effort and fired harmlessly wide.
The Norwegian champions were experiencing some joy getting in behind the Arsenal backline, but Turner was always equal to the task, coming out of his area well to snuff out any danger.
Arsenal got to half time with their 1-0 lead intact, but Taylor was called into action again in the 52nd minute, getting down low to save from Pellegrino after Bodo/Glimt won the ball back in midfield.
Ola Solbakken and Runar Espejord both fired just over the bar in quick succession for the hosts, who began to turn the screw, albeit without that final killer ball.
Arteta did not hesitate to introduce Granit Xhaka and Gabriel Martinelli with less than an hour gone as Bodo/Glimt continued to threaten, and there was little in the way of attacking inspiration from those in black.
Bodo/Glimt had the ball in the back of the net following a last-minute corner, but the referee's whistle had long gone for a foul on Turner, and Arsenal held out for their sixth win on the bounce in all tournaments.
Arsenal continue to sit pretty at the top of Group A with nine points from three matches, and they now prepare to face Leeds United in Sunday's Premier League clash before meeting PSV Eindhoven in their rescheduled Emirates encounter next Thursday.
Should Arsenal overcome Ruud van Nistelrooy's side next week, they will be assured of a place in the knockout round playoffs at the very least.