Tigers struck the first blow almost immediately. An initial push from a line-out maul was repelled by the home defence, but when the ball was recycled, Cracknell was able to burrow under the bodies to score.
Glasgow responded well, banging at the Leicester door, only to be denied by some ferocious defence. When Warriors did find a way through, it was in slightly fortuitous circumstances.
There seemed to be a mix-up in the line-out call when Johnny Matthews throw sailed right over the top, but Adam Hastings was on to it and slipped a pass for Venter to crash over.
The home side were on top and from a dominant scrum on the Leicester put-in, Vailanu was able to peel off the base and blast over.
With a strong wind at their backs, Glasgow continued to dominate territory and possession and looked to add to their advantage, but Leicester stood firm in the face of all that pressure. On the occasions they were opened up, their scramble defence was superb.
Hastings knocked over a penalty to send Glasgow in 17-5 up at the break.
Turning around to play into the breeze in the second period, the concern for the home side was whether a 12-point cushion was enough, but they added to it in the opening moments of the half, Vailanu going over from close range for his second of the evening.
The breeze was proving no problem at all for the hosts as they moved through the gears, and Hastings finished off a sweeping move for a try number four.
It was five almost immediately as Warriors sprung again straight from the restart, Kyle Rowe breaking free and finding Horne running a trademark support line to go in under the sticks.
This was turning into a bit if a doing, though Kata stemmed the tide somewhat by ghosting through to score Leicester's second try of the evening.
They should have scored again when Freddie Steward broke through and fed Dan Kelly, but with the line at his mercy, the centre inexplicably fumbled forward and the chance was lost.
Tigers third try did come eventually from Liebenberg, but by then the mountain to climb was too high for the visitors.
Venter powered over for his second try of the evening to round off a dominant Warriors performance.
Next up for the URC champions is one of European rugby's toughest tests, a last-eight showdown in Dublin with a Leinster side fresh from a 10-try demolition of Harlequins.
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