Glasgow Warriors made history as they overcame Lions in the Challenge Cup to reach their first European semi-final.
Franco Smith's men, who visit Scarlets in the last four, made a fast start as Jack Dempsey and Jamie Dobie crossed.
Sanele Nohamba halved the deficit after the interval but Zander Fagerson restored Glasgow's 14-point advantage.
Lions responded again, through Francke Horn, but a George Horne penalty and Tom Jordan try provided breathing space before Morne Brandon's late score.
The semi-final will be played on the final weekend in April.
Glasgow's start was blistering, a continuation of recent demolition jobs, all pace and power and ruthless finishing. Lions had the misfortune to lose their wing Rabz Maxwane after just seven minutes, a sickening collision with Matt Fagerson removing him from the game on a stretcher.
That was one setback and there was another in its wake when Glasgow scored off a training ground line-out move. It was a peach of a move, ball off the top from Richie Gray, Rory Darge and Horne then helping it on to Sione Tuipulotu on the crash ball, a pivot from the centre and an offload to the incoming Dempsey and over the number eight went.
The speed of it was the thing. The timing, too. Sheer class. The Lions suffered another loss shortly after when their flanker Ruan Venter departed.
No sooner had he gone than Glasgow struck again. Again it was all about accuracy and rapid pace, speed of hands out the line until Ollie Smith put Dobie away. Just as he had done with Dempsey's score, Horne put over the conversion. The Warriors led 14-0 and it was oh so easy.
And then it wasn't. For much of the rest of the half, and into the early minutes of the second period, the Lions dominated territory and field position. Their big carriers - nobody bigger, in every sense, than their tighthead Asenathi Ntlabakanye who finished with an astonishing 22 carries - came into the game in a major way.
They had multiple opportunities to crack the Glasgow defence - line-out, scrums, a tap penalty all deep inside the home 22 - but couldn't get the job done.
Glasgow's defence was as outstanding as the Lions line-out was suspect (Glasgow's own line-out would start to buckle later on). They lost a load of ball out of touch. The ball they won was messy more often than not, Gray was a total nightmare for them. Glasgow's aggression meant the visitors ended the half without a score, which must have infuriated them.
They put that right within a few minutes of the restart and for a Glasgow team that had defended their own line with a thunderous zeal, it was a soft score to give up.
Nohamba tapped and went from a penalty inside Glasgow's 22 and he ran straight to the posts, then converted his own try to remind the hosts that this was not going to be as straightforward as they might have thought.
Those were concerning moments from Glasgow because they will have known too well what the Lions are capable of when their tails are up.
In last week's last-16 match with Racing they were 15-7 ahead when they suffered a red card. They motored on to win 51-28 with a one-man disadvantage.
Glasgow needed to stamp out their mini-revival - and they looked to have done it when Gray pinched a line-out deep in the Lions half, the catalyst for pressure being piled on and Zander Fagerson piling over. With the conversion, the 14-point cushion was restored.
Then 14 became seven again when Horn went over. The conversion was good and Scotstoun started to fret once more.
The Glasgow Horne eased the gap out to 10 with a penalty and the nerves evaporated entirely when Huw Jones went through a gap, linked with Smith, who set Jordan on his way.
The unerring Horne lashed over the conversion. There was another response from the Lions when Brandon went over in the dying minutes, but Glasgow saw it out.
A first European semi-final, but they won't settle for that. They have the Scarlets in Llanelli next - one of the biggest in their history.
'Fantastic effort in defence' - reaction
Glasgow head coach Franco Smith: "It was an interesting performance in the sense that we didn't have the ball much.
"We looked nervous in attack which is uncommon, but I can understand why - there's a lot of passion and the guys don't want to disappoint. There was a fantastic effort in defence. It's fantastic for the club and I'm pleased to be a part of it.
"[Scarlets] are playing well at the moment. Every game is important now, so we will take things one game at a time."
Glasgow: Smith; Steyn, Jones, Tuipulotu, Dobie; Miotti, Horne; Bhatti, Turner, Z Fagerson, Cummings, Gray, M Fagerson, Darge, Dempsey.
Replacements: Matthews, McBeth, Berghan, Du Preez, Bean, Gordon, Price, Jordan.
Lions: Horn; Maxwane, Rass, Louw, Van der Merwe; Lombard, Nohamba; Smith, Botha, Ntlabakanye, Alberts, Schoeman, Kriel, Venter, Horn.
Replacements: Brandon, Rijnsburger, Dreyer, Delport, Gordon, Van den Berg, Jonker, Coetzee.