Teenager Jamie Murphy netted an 18-minute hat-trick and Ross Forbes, Shaun Hutchinson, Paul Slane and Robert McHugh all grabbed their first career goals as Well emphatically banished memories of their 1-0 second qualifying round first leg defeat in Albania last week, going through 8-2 on aggregate.
Jim Gannon's side scored seven goals in the space of 34 minutes and then added one more, thus comfortably securing a third qualifying round clash with Steaua Bucharest.
The stunning victory vindicated a major reshuffle by Gannon, who handed debuts to Steve Jennings and Chris Humphrey and gave Hutchinson his second appearance in central defence.
Full-backs Steven Saunders and Stevie Hammell were dropped with Mark Reynolds and Jim O'Brien moving into their places and Murphy brought in up front.
Motherwell had enjoyed plenty of possession in Albania and John Sutton missed a penalty, but the 38C heat blunted their attacking edge. There was no such problem in the cool Lanarkshire breeze at Airdrie's Excelsior Stadium.
There were goals in the air from the start with Murphy and Devis Mema forcing saves at either end in the opening stages.
Flamurtari adopted an attacking approach but their frailties at the back were soon evident when Humphrey blazed over after Sutton headed on Stephen Craigan's simple long ball.
Murphy and Jennings were both denied in a frantic goalmouth scramble after Shpetim Mocka fumbled Forbes' free-kick but the Steelmen soon broke the deadlock in the 16th minute.
Mocka, who had looked solid in the first leg but barely caught anything all night, parried Sutton's header from a Slane corner and Murphy forced home.
Well took the lead in the tie three minutes later after the visitors were caught woefully short at the back when their corner was cleared.
The hosts had a three-on-one advantage when Forbes sent Slane clear from inside his own half and Murphy tucked home the rebound after the winger's shot hit the post.
Slane had just six minutes to wait for his goal when he cut inside and fired just inside the post from 25 yards.
Forbes sent Mocka the wrong way from a 28th-minute penalty after Bledion Guga clumsily pushed Sutton and Murphy completed his treble in the 34th minute.
Artan Sakaj gifted possession to Sutton when he fell over in the centre circle and the former St Mirren man fed the 19-year-old, whose low drive from 20 yards slid through Mocka's legs.
Hutchinson diverted a Forbes free-kick just inside the post in the 38th minute and Forbes rattled home his second from 20 yards following Sutton's lay-off in the 50th minute.
Flamurtari sub Elham Galica should given some comfort to the 50 away fans following a mistake by Craigan, but he slipped the ball wide from 12 yards.
At this point Motherwell looked like they could score every time they went forward and Humphrey, Hutchinson and Sutton all missed half-chances.
But the visitors grabbed a 64th-minute consolation when Odise Roshi ran on to a simple through ball and slotted past Michael Fraser.
Well had lost their impetus and Gannon brought on Keith Lasley and McHugh for new signings Jennings and Humphrey.
McHugh continued the goal spree in the 72nd minute when he rolled the ball into an empty net after Mocka had dropped a high ball, and Sutton missed a great chance to join the scoresheet when he headed wide from six yards.
Gannon now continues the eventful start to his Motherwell reign in Romania next Thursday.