The Magpies - who hit the woodwork twice - were on top for large spells at a raucous St. James' Park, but Ødegaard's low drive put the visitors in front in the first half.
United had their chances, and goalkeeper Nick Pope played his part by repelling the title-chasing Gunners with a string of important saves, but Schär's own goal with 20 minutes remaining ended their hopes of mounting a fightback.
It was a breathless first period on Tyneside. With only 70 seconds gone, it looked as though United would move ahead. Joe Willock's burst from the middle of the park took him past his man and down the left and his cross found Jacob Murphy at the far post. The winger took a touch and curled a shot goalwards, but it rebounded off the foot of the post.
The noise inside St. James' was deafening. Eight minutes in, the hosts thought they had a penalty when Bruno Guimarães' effort was blocked by Jakub Kiwior. Referee Chris Kavanagh pointed to the spot immediately but a protracted VAR assessement cost the Magpies; after another look, Kavanagh appeared to decide that there wasn't enough in it for a spot kick to be given.
With just under quarter of an hour gone, the Gunners - who were on the back foot in the early stages - struck, and it was somewhat opportunistic. They had offered little going forwards up that point but when a free kick was cleared, they recycled possession well, with the ball finding Ødegaard outside the box. His strike was clean and low and flashed through a crowd of bodies before nestling in the bottom right-hand corner.
The goal brought Arsenal to life and but for Pope, they could have gone further ahead within minutes. Ødegaard's through ball gave Gabriel Martinelli a chance which Newcastle's goalkeeper did well to block before the visitors' goalscorer had a go himself, forcing Pope to turn it round the post at full stretch.
Bukayo Saka then found a gap in the Magpies' backline and ran through to sting Pope's palms and, despite an Aaron Ramsdale save to deny former Arsenal man Willock as United pushed, Mikel Arteta's side finished the half on top. It took another Pope save to keep the score at 1-0 in the final seconds of stoppage time after Ødegaard sidetepped his marker in the box and looked to roll home a second.
United came out fighting. Murphy's run down the right just after the restart ended with a teasing clipped cross which was met by Alexander Isak, whose downward header thumped against the base of the post. From a Kieran Trippier free kick moments later, Schär headed what he must have thought was the leveller - until Ramsdale threw an instinctive arm at it to repel the centre back's attempt.
Willock had a sight of goal just before the hour after a good move down the left, which began with Dan Burn's searching ball down the line. It was picked up by Isak, who cut inside and teed up his teammate, but Granit Xhaka got back just in time to slide in and prevent him from getting his shot away.
Arsenal's attacks seemed to be less frequent after the break but one of them almost resulted in a goal - Martinelli's delicate effort leaving Pope helpless as it struck the crossbar. It was proving to be a tricky afternoon for another Brazilian, Bruno Guimarães, who saw a tame sidefooted shot saved by Ramsdale, and with 20 minutes remaining Newcastle's task became even harder.
The Gunners' second of the day came thanks to a quick breakaway, led by Martinelli. He got to the byline and cut the ball back in hope, but the centre cannoned off Schär's outstretched leg and flew past Pope from close range.
Howe sent on the fit-again Allan Saint-Maximin and Anthony Gordon late on, but to no avail. Schär hooked one over the bar from distance and Saint-Maximin saw his effort tipped wide by the England stopper in a late rally from the hosts, but it proved fruitless as Arteta's men held firm to take the points.
