Glasgow secured a first European final with a thrilling Challenge Cup last-four win over Scarlets in Llanelli.
George Horne scored 15 points, with two tries from Stafford McDowall and further scores for Johnny Matthews and Rory Darge.
Scarlets responded with a Steff Evans try and four Sam Costelow penalties, but were hampered by a series of injuries during a brutal battle.
Glasgow will face the winners of Toulon and Benetton on 19 May in Dublin.
Franco Smith's side are the first Scottish team to contest a European final since Edinburgh lost to Gloucester in the Challenge Cup in 2015.
Glasgow are also still battling on two fronts, with a United Rugby Championship (URC) home match against Munster next weekend.
Defeat for the Scarlets represented a fifth European semi-final loss, with the four previous defeats coming in the Heineken Champions Cup.
Dwayne Peel's side battled manfully, with captain Josh Macleod again outstanding, but struggled to cope with losing key men Johnny Williams and Vaea Fifita to injury.
Scarlets can also point to a controversial decision by French referee Mathieu Raynal not to award a 68th-minute penalty try and yellow card for a deliberate knock-on.
A decision in their favour would have brought the hosts back into the game, but Glasgow fully deserved the victory with five tries to one.
The Scarlets' loss also ended the season for the Welsh regions with no sides having qualified for the URC play-offs.
Near-capacity crowd
It has been a disastrous season for Welsh rugby so to have Scarlets in a European semi-final in front of 13,000 fans in Llanelli was a welcome respite.
Retiring Wales flanker Aaron Shingler was denied a home farewell after failing to recover from a back injury, but was presented to the Scarlets crowd before the game.
Scotland lock Richie Gray was due to make his 100th Glasgow appearance, but was a late withdrawal with JP du Preez coming into the starting side.
Glasgow, who came into the match having only lost once in their last 17 games, ignored the disruption and home sentiment with their strong start.
A typical sweeping attacking move started by incisive breaks from Tuipulotu, captain Kyle Steyn and player-of-the-match Jack Dempsey gave immediate warning of their intent.
That laid the platform for powerful centre McDowall, who was filling the midfield void left by Huw Jones, to score, with scrum-half Horne converting.
Scarlets suffered two early injury blows with Scotland prop Javan Sebastian and Wales centre Williams forced off injured with Sam Wainwright and Ioan Nicholas coming on.
It proved a chaotic opening quarter with both sides looking to play from deep, producing continual turnovers and numerous handling errors.
This resulted in a series of scrums with Glasgow prop Zander Fagerson having the upper hand over his 2021 British and Irish Lions team-mate Wyn Jones.
While Glasgow were edging the scrums, Scarlets were initially dominating the breakdown battle with flankers Macleod and Dan Davis successful in turning over possession to halt the visitors' momentum.
Costelow reduced the deficit to one point with two penalties before some lacklustre defence from McDowall put his side in trouble.
A wayward line-out gifted Scarlets a try chance with Sam Lousi's flick pass allowing full-back Johnny McNicholl, who had been preferred to Leigh Halfpenny, to release Evans, who cantered over unopposed to give the home side the lead.
Lock Morgan Jones became a third Scarlets casualty of the first half after he came off second best in a collision with Zander Fagerson.
This prompted a forwards reshuffle with number eight Carwyn Tuipulotu coming on and Fifita moving up to lock after he had started in the back row in place of the injured Sione Kalamafoni.
This disruption failed to halt the Scarlets' revival as Costelow slotted over a penalty just before half-time to give the hosts a 14-7 half-time lead.
Glasgow's second-half power tells
Glasgow responded at the start of the second half as locks Du Preez and Scott Cummings combined to release Horne, who converted his own try to level the scores.
In the build-up to the try, the television match official (TMO) spotted a high tackle from Wainwright on Darge which resulted in a yellow card for the Scarlets prop.
Costelow restored the advantage with his fourth penalty before Scarlets' injury curse continued as Fifita was forced off.
That prompted the introduction of flanker Iestyn Rees with Lousi the only specialist Scarlets second-row remaining.
Glasgow took advantage as replacement hooker Matthews, who scored five tries against Dragons earlier in the competition, was driven over with Horne again converting.
With the Scarlets line-out malfunctioning, Glasgow's power game told again with flanker Darge powering over for Glasgow's fourth try.
French referee Raynal angered the Scarlets crowd by failing to give a penalty, yellow card and penalty try to the home side after full-back Ollie Smith had appeared to deliberately knock the ball on. Instead the officials simply gave a Scarlets scrum.
McDowell completed the scoring with his second try after a well-worked set-piece move as Glasgow now await the victors of Sunday's semi-final between Toulon and Benetton.
Scarlets head coach Dwayne Peel: "I thought at half-time it was an even contest and we were right in it. Glasgow started both halves well, they had a good maul, and we probably weren't quite as accurate as we could have been.
"I can't fault the effort of our boys. I thought we were right in the game. We had three head injury assessments (HIAs) in the first 20 minutes which is unheard of. To lose all three, and then to lose Vaea (Fifita) as well is difficult.
"That definitely hurt us later on in the game because they were bringing on fresh bodies, and we didn't manage to get the fresh legs on."
Glasgow head coach Franco Smith: "The Scarlets threw everything at us tonight, the crowd was immense. For us to handle the pressure we put on ourselves in the first half was fantastic.
"We felt a little bit nervous, even guys who have played Test-match rugby. Lesson learned, we went back to the process, and I'm looking forward to seeing what we take out of this review.
"Benetton have been playing well lately, have had some good away games in South Africa. Toulon are a consistent power and have some star-studded players. It's going to be interesting, we will be sitting watching."
Scarlets: McNicholl; Steff Evans, Roberts, J Williams, Conbeer; Costelow, G Davies; W Jones, Owens, Sebastian, M Jones, Lousi, Macleod (capt), D Davis, V Fifita.
Replacements: Shaun Evans, Mathias, Wainwright, C Tuipulotu, Rees, Hardy, D Jones, Nicholas.
Glasgow: Smith: Cancelliere, S Tuipulotu, McDowall,Steyn (capt); Jordan, G Horne; Bhatti, Turner, Z Fagerson, Cummings, JP du Preez, Darge, Vailanu, Dempsey.
Replacements: Matthews, McBeth, Berghan, Wilson, Bean, M Fagerson, Price, Weir.
Referee: Mathieu Raynal (France)
Assistant referees: Tual Trainini & Pierre Brousset (France)
TMO: Eric Gauzins (France)