But they got off to the worst possible start in front of a raucous Elland Road as Danny Welbeck gave Brighton the lead in the 21st minute.
Leeds fans grew more and more frustrated as the match wore on and an equaliser remained elusive, singing the name of previous coach Marcelo Bielsa and chanting "sack the board" in the direction of the club's owners in the director's box.
However, there was late, late drama as Struijk arrived at the back post to snatch a draw in the 92nd minute, sparking wild celebrations among players, staff and supporters.
It could be the goal that preserves Leeds' Premier League status next season, with Jesse Marsch's side now on 35 points ahead of next weekend's final round of fixtures, one clear of 18th-placed Burnley, who have two games left. Brighton stayed 10th.
"Our goal was to get three points today but we knew that any points and Burnley not getting any would shift the pressure a bit the other way," Marsch said.
"Burnley still have a match in hand but to fight that way and stay in the match was impressive. We were unlucky not to come away with more.
"This is a psychological battle we are in. We need to show up next week and show us at our best."
Leeds fans gave a rousing welcome to a host of former players from their 1974 and 1992 top-flight title-winning seasons before kick-off and the crowd were further buoyed by Burnley's loss being announced pre-game.
The visitors, already assured of their best-ever Premier League points return this season, soon settled into the match and created several openings, with Solly March dragging their best early chance wide.
The pressure told as the Premier League's second worst defence was again breached, with Welbeck muscling Leeds defender Diego Llorente off the ball before clipping a sumptuous finish into the net.
Brighton chose to sit back in the second half and invite pressure from Leeds, with Raphinha firing a thunderous free kick from distance towards goal but Sanchez was again at his best to deny the chance.
Often Leeds's most potent attacking threat, Raphinha almost nudged one home at the back post, only to see his attempt roll wide.
Leeds looked to be heading for a fourth straight loss until Struijk edged his header over the line.
The Yorkshire side have scored a Premier League high seven stoppage-time goals this season. Crucially, six of those seven goals won points.
"It is frustrating with the lateness of the goal but in fairness Rob [Sanchez] has had to make some saves and we needed a second goal to kill it off with the situation Leeds are in," Brighton boss Graham Potter said.
"While it is 1-0 the game is alive and Leeds created chances in the second half. I am not too despondent with a point."