The Gunners, who went into the match with a 100% record in this season's competition, were unable to find the breakthrough against their Serbian visitors, who came closest to scoring when they hit the crossbar in the first half.
Indeed, Red Star - who move up to second in Group H courtesy of the point - created the best chances of the match as a youthful Arsenal side saw the club's perfect home record across all competitions this season come to an end.
BATE Borisov's 5-2 defeat to bottom club Koln means that a point is enough to guarantee Arsenal's place in the knockout rounds, though, while also seeing Red Star move into the qualifying positions with two matches left to play.
Arsene Wenger made a full 11 changes to his starting lineup with one eye on Sunday's trip to Premier League leaders Manchester City, so it was no great surprise to see the Gunners struggle to get into any sort of rhythm in the first half.
The first big chance of the contest fell to Olivier Giroud after 10 minutes following good work from Ainsley Maitland-Niles, but the France international - who scored the only goal of the reverse fixture a fortnight ago - was denied by Milan Borjan.
Such openings were a rarity in a first half short of quality in the final third, though, and it wasn't until 10 minutes before the interval that either side created another chance of note when Giroud was again thwarted by the Red Star keeper.
At the other end, Matt Macey was making only his second Arsenal start and the first European appearance of his career, and he was called into action shortly afterwards to turn a long-range drive from Slavoljub Srnic over the crossbar.
The resulting corner was played short to Nemanja Radonjic whose strike was deflected, but Arsenal failed to react to the looping rebound into the area and it took an athletic stop from Macey to turn Vujadin Savic's subsequent header onto the crossbar as Red Star came inches away from breaking the deadlock.
The visitors created arguably an even better chance in the final minute of the first half when Richmond Boakye raced clean through on goal, only to fluff his lines and fire a poor finish wide with only the keeper to beat.
It was symptomatic of the lack of cutting edge posed by either side, and not much changed in the second half with Giroud blasting an early strike over the crossbar from 25 yards.
Macey was then called into action again to deny Srnic's strike from a tight angle before the Gunners saw two penalty appeals turned down in a matter of seconds - both against Savic as first Jack Wilshere went down under the defender's challenge before the ball bounced up against his arm.
Wilshere was enjoying a growing influence on the match and must have thought he had broken the deadlock shortly after the hour mark when he took the ball off the feet of his teammate inside the area before clipping his finish past the keeper, only for Damien Le Tallec to produce a stunning clearance off the line.
Giroud flashed another effort past the post moments later before Theo Walcott came close in the final 15 minutes, steering a header from Wilshere's cross narrowly wide of the target.
Red Star continued to create chances of their own, though, and Boakye spurned another glorious opening with 10 minutes remaining when he powered an uncultured finish wide of the near post having been played through after Mathieu Debuchy had gone down off the ball.
There was to be no breakthrough for either side in the end, although that worked out well for both as Arsenal's qualification was confirmed with two games to spare, while Red Star now sit in pole position to join them in the last 32 from Group H.