A try from Martin Castrogiovanni and eight points from the boot of Toby Flood gave the Tigers a slender 13-10 half-time advantage, with Morgan Stoddart touching down early on for the Welshmen.
But when the teams resumed after the break Leicester cut loose, Ben Youngs, Thomas Waldrom and Tom Croft all going over in a frantic 10-minute spell, before Croft and Matty Smith grabbed further tries to earn a bonus point.
The emphatic win left Premiership champions Leicester with 10 points from two matches, four clear of second-placed Perpignan, who beat Benetton Treviso 35-14 in France. Scarlets lie third on five points.
“I felt we needed more points out of the first half. At half-time they were very much in the game,” said Tigers director of rugby Richard Cockerill. “They took their try well and we’d have liked to have scored a couple more of our own but we didn’t take our opportunities. But we kept the intensity and composure and those five tries in the second half included some quality bits of rugby.”
In Pool Four, Biarritz kept up their 100 per cent record with a bonus-point 35-15 triumph at home to Ulster. Wing Takudzwa Ngwenya scored two tries for last season’s runners-up, while scrum-half Dimitri Yachvili contributed 15 points.
Elsewhere, Leeds Carnegie lost 13-22 at home to Stade Francais as the French side held onto top spot in Pool Four of the Amlin Challenge Cup.