Alejandro Garnacho thought he had won it in the 88th minute, but his goal was ruled out for a narrow offside. Minutes later, in the 97th minute, Rice collected a Bukayo Saka corner to beat Andre Onana at his near post.
As United pushed for an equaliser, Jesus ran through on the counter to settle the game.
Marcus Rashford gave United the lead against the run of play in the 27th minute, lashing home a shot after a counter-attack, but their joy did not last long though as Martin Odegaard fired Arsenal's equaliser less than a minute later.
Arsenal were awarded a penalty midway through the second period when Kai Havertz went down in the area but referee Anthony Taylor change his mind after checking a monitor.
Mikel Arteta's side have 10 points from four games with United on six after already losing twice.
Additional reporting by Reuters.