The former England youth coach took over at the Liberty Stadium in the summer, inheriting a squad who finished 10th last season and lost star winger Daniel James and prolific striker Oli McBurnie to the Premier League.
Cooper has fitted in seamlessly and, having soaked up plenty of pressure and seen Leeds miss plenty of chances, substitute Routledge's 90th-minute strike took the Swans two points clear at the summit.
Both teams came into the top-of-the-table showdown unbeaten with 13 points from five games, with Leeds only top on goal difference, but it was United who slipped to their first league defeat of the campaign.
The hosts totally dominated yet, like on numerous times last season during their unsuccessful promotion drive, their finishing let them down.
The Welsh side struggled to contain Leeds early on and Mike Van Der Hoorn, making his 100th appearance for the club, almost turned a stray Stuart Dallas cross beyond Freddie Woodman and into his own net.
Gjanni Alioski had his two chances at the back post, firing over when unmarked and then acrobatically hooking an effort at Woodman.
Patrick Bamford then headed off target twice, the first when free at the front post, before Swansea showed signs of stirring.
Leeds almost took the lead on the stroke of half-time through fit-again captain Liam Cooper, who nodded a corner against the frame of the goal and then hooked the rebound over.
Marcelo Bielsa's side started the second half how they finished the first.
Bamford sliced an effort wide and Woodman saved a Pablo Hernandez drive from the edge of the box.
The further missed chances saw Bielsa introduce striker Eddie Nketiah on the hour mark.
Chants of "Eddie, Eddie Eddie" rang around the ground and the on-loan Arsenal striker almost made an immediate impact, poking over the top after he latched onto a through ball.
Hernandez opted to take too many touches rather than shoot when unmarked at the back post, then blazed a shot just over as Leeds piled on the pressure.
Nketiah almost scored late on, heading wide from Dallas' cross when under pressure from Woodman, but it was Swansea who won it at the death.
Leeds failed to clear a corner numerous times and Routledge fired into the bottom corner.