It was the second time this season that the league's leaders had dropped points.
Saints have not lost in the league since the beginning of the season and have a three- match winning streak.
The first goal of the game was scored by Granit Xhaka, who swept the ball into the roof of the net from Ben White's low delivery.
Gabriel Martinelli had a shot picures wide and Gabriel Jesus was denied by the goalkeeper as the visitors threatened to double their lead before the break.
Jesus, who had broken clear of the Saints defence, was denied a goal by a last-ditch challenge from Elyounoussi.
The Norwegian's effort was ruled out after the ball had gone out of play in the build up to the game.
The result leaves Mikel Arteta's team two points clear of Manchester City, while the Saints are three points away from the bottom of the table.
After taking an early lead on the south coast, they appeared to be on their way to a 10th league victory of the season.
The Switzerland international made no mistake from White's accurate right-wing cross after Odegaard missed a shot from Xhaka's pass in the 10th minute.
After opening the scoring,Arsenal continued to pose the greater threat in the final third and were close to adding to their lead through Martinelli and Jesus.
Jesus' failure to get his shot off before Elyounoussi's goal-saving challenge proved costly as Saints claimed a deserved share of the spoils.
There were no complaints from the visiting players when Odegaard's powerful finish from the cut-back was ruled out, with replays showing the ball had gone behind before the Scotland full-back could catch it.
It is the first time in 28 matches that the Gunners have failed to win a match.
Lyanco's long-range effort and a flurry of corners towards the end of the half weren't enough to trouble Ramsdale before the break.
A switch from a back four to a back five around the 20 minute mark helped stem the flow of attacks from the visitors.
Clear-cut opportunities were few and far between in the second half, but they were not the only ones.
The Scotland player calmly slid the ball past the onrushing Ramsdale after superb play from Elyounoussi to pick out his teammate on the edge of the penalty area.
The scrutiny on Hasenhuttl had increased after the Manchester City defeat, but a third successive game without a loss has helped ease that.
Only Crystal Palace have collected more points after falling behind this season.
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