After playing out an enthralling 2-2 draw in the first leg at Ibrox, the Gers secured a 1-0 victory in an equally entertaining encounter in the reverse fixture at the Philips Stadion on Wednesday night, with summer signing Antonio Colak scoring the only goal on the hour mark.
Rangers will now join Glaswegian rivals Celtic in Pot 4 for Thursday's Champions League group-stage draw at 17:00pm (BST) in Istanbul, while PSV will drop down into the Europa League.
Ruud van Nistelrooy's side made a bright start to the contest, but Rangers - who decided to leave out star striker Alfredo Morelos - responded well and enjoyed a dominant 15-to-20 minute spell midway through the first half.
The best first-half chance for the visitors fell to John Lundstram in the 26th minute when he placed a side-footed strike from the edge of the penalty area narrowly wide.
After that, PSV finished the first half strongly and pressed hard for the opener in the final 10 minutes, with Giovanni van Bronckhorst's men desperate to hear the half-time whistle.
Rangers conceded from two corners in the first leg, and they were fortunate not to concede from another in the 38th minute when Ibrahim Sangare - who scored at Ibrox last week - found himself unmarked in the penalty box before watching his header glide over the crossbar.
Jordan Teze threatened down the right flank and picked out captain Luuk de Jong in the 42nd minute, but the latter could only direct his stretching effort straight at goalkeeper Jon McLaughlin.
Manchester United target Cody Gakpo then fired a glorious chance over the crossbar just moments later after bursting past a few Rangers defenders on the edge of the area.
PSV were dealt a blow for the second half as De Jong was forced off at the break due to injury, with midfielder Xavi Simons - who had previously been linked with a move to Ibrox - coming on as his replacement.
Both sides made a lively start after the break, but it was Rangers who almost made the breakthrough in the 55th minute when Tom Lawrence saw a curling right-footed strike hit the underside of the crossbar.
However, Rangers were gifted the opening goal just five minutes later when Malik Tillman pressed high and caught centre-back Andre Ramalho in possession on the edge of PSV's penalty box, before showing great composure to set up Antonio Colak for a close-range tap in.
PSV stopper Walter Benitez did well to keep out a drilled effort from Tillman in the 66th minute, before Phillip Max and Gakpo came close at the other end moments later, the latter kept out by a brilliant save from McLaughlin.
The hosts tried desperately to push forward in the closing stages, but frustrations grew and the game became scrappy which played into Rangers' hands.
The Gers had won only one of their last 11 away matches in European competition heading into Wednesday's second leg, but they managed to hold on for the victory and earn their place back in the big time.
Rangers will now turn their attention to Saturday's Scottish Premiership clash at home against Ross County, while PSV will make the trip to Excelsior for their Eredivisie encounter on Sunday.