Saka scored midway through the first half in Norway to hand the Gunners their third win in as many games in Group A.
“I’m really happy to beat a team who have won their last 14 games here at home,” manager Mikel Arteta told BT Sport.
“We were expecting a really tough night here and we still made a lot of changes, but I’m glad we found a way to win.”
Arsenal are two points clear of PSV Eindhoven, who beat FC Zurich 5-0. Bodo/Glimt have four points in four games.