Joe Willock's deflected strike looped over a stranded Kevin Trapp in the first half before late goals from Saka and Pierre-Emerick Aubameyang handed the Gunners all three points in their first game of this season's competition.
Dominik Kohr was sent off in the 79th minute for a second yellow card after he caught the lively Saka's heels, as Arsenal surged to the top of Group F.
The first half was an open affair, with Saka - one of a number of youngsters starting the game - playing Lucas Torreira into the box inside the first five minutes, but the Uruguayan blazed over the bar when he should have scored.
Bas Dost also came close to punishing a hesitant Gunners defence, lashing wide of the top-left corner from inside the box and heading past the post as he caused problems with his height and physicality.
However, Arsenal had the better chances, with Willock firing wide and Smith Rowe denied when one-on-one, before the former finally broke the deadlock.
Saka played the central midfielder in down the left and he charged into the box before cutting inside and seeing his shot deflected over Trapp and in off the crossbar.
Granit Xhaka also rattled the bar early in the second half when his cross from a free-kick on the right sailed over everyone and pinged back off the woodwork.
Danny da Costa let Arsenal off the hook as he smashed his shot over the top, with the hosts' profligacy in front of goal costing them dearly, although Emiliano Martinez, making his first appearance of the season for the Gunners, proved himself a calming, reliable presence in goal throughout the night.
Frankfurt midfielder Kohr picked up a second booking in the 79th minute for a tired challenge on Saka, who had drifted past him, and the hosts collapsed after going down to 10 men.
Aubameyang produced a glaring miss in the 83rd minute as the onslaught began, somehow diverting Chambers's low cross wide at the back post with the goal gaping, but it would not matter.
Two minutes later, Martinez set Saka on his way to score the Gunners' second goal of the night, curled beautifully into the corner from outside the box after a pass from substitute Nicolas Pepe.
In the 87th minute, Saka grabbed his second assist of the night, setting Aubameyang up to finish low into the corner as Frankfurt pushed high up the field in desperation and left themselves exposed.
The result ends Arsenal's woeful record in Germany, having lost their previous four European away games against German opponents.
Unai Emery will now shift his focus to Arsenal's Premier League game on Sunday as they host newly-promoted Aston Villa at the Emirates Stadium.
EINTRACHT FRANKFURT (3-4-1-2): Trapp; Abraham, Hasebe, Hinteregger; Da Costa (Chandler 74'), Kohr, Sow, Kostic; Kamada; Dost (Paciencia 66'), Silva
ARSENAL (4-3-3): Martinez; Chambers, Mustafi, Luiz, Kolasinac (Maitland-Niles 80'); Torreira, Xhaka, Willock (Ceballos 72'); Smith Rowe (Pepe 60'), Aubameyang, Saka