Lowry's penalty and a late Quinn header saw the Premiership leaders exit while Joel Cooper bagged a double in Glenavon's 5-0 hammering of Portadown.
Jamie McGonigle and Eoin Bradley also netted twice as Coleraine thumped Ballymena in another one-sided game.
Dungannon Swifts progressed thanks to Douglas Wilson's superb extra-time strike in a 2-1 win over Warrenpoint.
Aaron Traynor's header put the the Championship side in front at Stangmore Park but the Swifts levelled on 73 minutes courtesy of a Jordan Dane own goal.
Wilson won it the first period of extra-time, his long-range shot whistling past Town keeper Jonathan Parr.
It was a close contest at Seaview with few clear-cut chances in a scrappy game between the league's top two.
That was until the 72th minute opener with Lowry firing home from 12 yards after the impressive Paul Smyth was brought down by BJ Burns.
Jamie Mulgrew showed his class for the second goal two minutes from time, beating two players and delivering a pinpoint cross for Quinn to nod in.
It was Linfield's third straight win over the Crues, who lead the title race by seven points.
It was all too easy for the Lurgan Blues against their mid-Ulster derby rivals Portadown at Shamrock Park.
Ports keeper Jack Duffin saved a Greg Moorhouse penalty but the Glenavon goals soon flowed, staring with a fine Ciaran Martyn strike.
Cooper jinked into the area before sliding between Duffin's legs on the hour mark and the forward's wonderful volley made it 3-0.
James Singleton slammed home from close range and Portadown's miserable afternoon was completed by a looping Mark Sykes header.
Coleraine cruised a derby win over the Sky Blues with Jamie McGonigle scoring at the second attempt for a 12th minute opener.
Darren McCauley saw his penalty saved by United keeper Ross Glendinning before Bradley struck with goals either side of the break.
The striker lifted the ball over the advancing Gledinning to double Coleraine's lead and then pounced on the keeper's error to net.
McGonigle secured his double as the Bannsiders marched into the semi-finals.
Stephen Lowry and Niall Quinn were on target as Linfield beat Crusaders 2-0 in the Irish Cup quarter-final.