Tries from Sam Johnson, Tommy Seymour, Matt Fagerson, Zander Fagerson and George Horne saw them to a fine away bonus-point success.
Leinster’s cause wasn’t helped early on by fly-half Ross Byrne being yellow carded after only three minutes for tackling Seymour without the ball, Adam Hastings sending over the resulting penalty for a 3-0 lead for the Warriors. The fly-half would go on to finish with 14 points.
The Irish did respond five minutes later when Dave Kearney crossed but Joe Tomane was wayward with the extras, meaning they led by two.
Hastings would nudge his side back in front before a wonderful break from Ali Price on his own 22 led to Johnson fending en route to a try. With Hastings’ two the Warriors had raced into a 13-5 lead in Dublin and were looking in good shape with a quarter of the game gone.
Leinster managed to score next when sustained pressure in the Glasgow 22 led to Rob Kearney crossing. However, the half would end with Glasgow grabbing their second try of the day as a wonderfully timed pass from Hastings to right wing Seymour saw him make it 18-12.
Rob Kearney would get his second 10 minutes into the second period after another multi-phase play from Leinster which took them ahead.
The family affair continued when brother, Dave, also picked up his brace but Glasgow did respond via their pack as Matt Fagerson crossed for the lead, which was built on soon after as Zander Fagerson powered over the whitewash. Suddenly Glasgow were in front, 32-24.
And the icing was then added by replacement Horne, whose intercept crossing made it a five-try victory for Dave Rennie’s men on the road.
What a finish at the Sportsground 🤯@cardiff_blues gave themselves a chance with two late tries, but @connachtrugby managed to hold on and make the Final Series 👊#GUINNESSPRO14 #CONvCBL pic.twitter.com/JdFwGxrHVl
— PRO14 RUGBY (@PRO14Official) April 13, 2019
Caolin Blade, Gavin Thornbury, Jack Carty and Matt Healy crossed the try-line while Carty also added nine points off the tee in their win.
For the Blues this result ends their hopes of knockout rugby as they remain in fifth position in Conference A with one round remaining.
Meanwhile, the Scarlets have an outside chance of reaching the end of season play-offs after thrashing Zebre 42-0. Wayne Pivac’s men have endured a disappointing campaign but they were too good for the Italians as Uzair Cassiem (twice), Hadleigh Parkes, Johnny McNicholl, Kieran Hardy and Ioan Nicholas all touched down, leaving them fifth in Conference B, two points behind third place Benetton.
In the final game of Round 20, the Cheetahs were too good for the Dragons after they sealed a 38-13 triumph over the Welsh region. The teams were level going into the final 25 minutes as Henco Venter and Taine Basham traded tries, but the South Africans pulled away in the last quarter via Shaun Venter, Tian Schoeman, Louis Fouche and Rabz Maxwane efforts.