Results earlier in the day meant that only a win would be enough for Unai Emery's side, but Glenn Murray's second-half penalty cancelled out Pierre-Emerick Aubameyang's opener - which also came from the spot - at the Emirates Stadium.
The Gunners are not yet mathematically out of the running with three points now separating them from fourth-placed Tottenham Hotspur, but they would need to win, Spurs to lose and an eight-goal swing in goal difference on the final day to leapfrog their local rivals.
Instead Arsenal's hopes of qualifying for next season's Champions League now look reliant on them winning the Europa League, where they already have one foot in the final courtesy of their 3-1 win over Valencia in the first leg of the semi-final in midweek.
Arsenal had seen a couple of results go their way earlier in the weekend, with Manchester United's draw against Huddersfield Town ruling them out of top-four contention and Tottenham's defeat to Bournemouth ensuring that they were still catchable for their North London rivals.
However, Chelsea's win over Watford earlier on the Sunday meant that only victory would do for the Gunners, and they almost broke the deadlock inside two minutes when Henrikh Mkhitaryan fired a low strike through a crowd of bodies and against the inside of the post from just outside the area.
The hosts would not have to wait long for the opener, though, as referee Anthony Taylor pointed to the spot after ruling that Alireza Jahanbakhsh had brought down Nacho Monreal just inside the box after only eight minutes.
Replays showed that Brighton's record signing actually got a foot on the ball, but Aubameyang made no mistake with his penalty, sending the keeper the wrong way and tucking it into the bottom corner for his 20th league goal of the season.
Brighton refused to lie down despite their Premier League status being secured for another season courtesy of Cardiff City's defeat to Crystal Palace on Saturday, and Murray drew the first save from Bernd Leno after the Arsenal keeper had invited pressure with a poor pass out from the back.
Solly March then fired a shot into the side-netting having worked space inside the area, but Arsenal soon regained control of the game and it took a smart save from Mathew Ryan to keep out Shkodran Mustafi's diving header at the near post.
Ryan then denied Aubameyang a second goal by getting down quickly to push his well-controlled shot away, and the Brighton keeper was called into action again on the stroke of half time when Mkhitaryan tried his luck with a powerful strike from range.
It was Brighton that made the better start to the second half after Chris Hughton brought on Anthony Knockaert in place of Jahanbakhsh, and March drew a low stop from Leno six minutes after the restart having played a slick one-two with Murray.
March was heavily involved for the equaliser too, winning a penalty on the hour mark when he burst past Granit Xhaka down the left before having his heel clipped by the Arsenal midfielder inside the area.
Anthony Taylor was quicker to point to the spot on that occasion, and Murray fired home from 12 yards for the 200th league goal of his career and one which left Arsenal facing the prospect of falling short in the top-four race with a game to spare.
Aubameyang volleyed one shot off target after being found direct from a corner and then drew a routine save from Ryan with a long-range strike, but it was an opening 16 minutes from time which the Gabon international really should have scored.
Mkhitaryan's cross from the left made it all the way through to the Arsenal goalscorer, who somehow fired wide from eight yards when the Emirates Stadium as a whole was waiting for the net to bulge.
Lacazette then saw a shot snuffed out after a lovely piece of skill inside the area as Arsenal began to ramp up the pressure, although Brighton still posed the odd threat at the other end and both Bernardo and Yves Bissouma came close with powerful long-range strikes.
Indeed, Brighton should have gone on to win the game when they squandered two glorious chances with five minutes left, first March seeing his diving header from point-blank range come back off the post and then Pascal Gross somehow missing the target completely from the follow-up when it looked easier to score.
Arsenal could not make the most of that let-off, though, as their late pressure came to nothing to leave them realistically unable to catch Spurs, while also guaranteeing Chelsea's place in the Champions League next season.
ARSENAL (4-3-1-2): Leno; Lichtsteiner (Kolasinac 77'), Mustafi, Sokratis, Monreal; Torreira, Xhaka (Iwobi 77'), Mkhitaryan (Guendouzi 77'); Ozil; Lacazette, Aubameyang
BRIGHTON (4-5-1): Ryan; Bruno, Duffy, Dunk, Bernardo; Jahanbakhsh (Knockaert 46'), Gross (Kayal 92'), Stephens, Bissouma, March; Murray (Andone 78')