And they nearly extended their advantage two minutes later when Johne Murphy raced after Vesty's grubber kick.
Mick O'Driscoll, Johne Murphy and Alan Quinlan all touched down while Warwick added a penalty and a conversion.
But researchers are investigating whether Johne's is linked to Crohn's disease in humans.
ECONOMIST: Another cow disease with a possible link to humans
Ireland's scrum creaked late on as the Scots were awarded a penalty try but Johne Murphy scored a late Irish try.
Ben Woods touched down within seconds of coming on as a replacement after and Johne Murphy then skipped through for another try.
It was thought that pasteurising milk could cut the risk of Johne's being transferred to humans, but this is now being questioned.
Contepomi missed with a penalty attempt but then tackled Johne Murphy as he crossed the line, doing just enough to push the centre into touch.
The Tigers dominated in difficult conditions but only secured a narrow half-time lead when Johne Murphy surged over after good work from Jordan Crane.
The government officially launched a Johne's control programme three years ago.
ECONOMIST: Another cow disease with a possible link to humans
Johne's Disease (paratuberculosis) is a debilitating incurable intestinal disease in cattle.
The symptoms of Johne's Disease are acute weight loss and diarrhoea.
Tommy Bowe scored an early second-half try but David Wallace and Johne Murphy touched down before O'Connell saw red for a swinging arm on Jonathan Thomas.
He follows fellow first-team squad members George Chuter, Martin Castrogiovanni, Jordan Crane, Marcos Ayerza, Johne Murphy, Ben Pienaar and Brett Deacon in committing his future to the club.
Murphy also converted his own try before being replaced and Vesty took over the kicking duties when he converted a brilliant try from Tom Varndell which Johne Murphy created.
Flood kicked the resulting penalty and minutes later added the conversion to a try from Matt Smith after Johne Murphy had produced a splendid run to set it up.
"When Johne missed his kick I was fully prepared for them to get theirs but I feel very sorry for the guys that missed, " said Tigers captain Geordan Murphy.
Johne Murphy finished off a sparkling move featuring fine passes from Geordan Murphy and Aaron Mauger before Geordan Murphy spun through two tackles to finish of a second sweeping move.
Abendanon cynically tripped Johne Murphy as he chipped ahead but the fit-again Moody marked his appearance from the bench by winning the race to touch down for a rapturously-received score.
Johne Murphy went close to scoring a Munster try before the break but the ball just beat him to touch after he had hacked on a chip through from Tomas O'Leary.
Italy full-back Andrea Marcato pushed an early penalty wide for the hosts before Johne Murphy sliced through in midfield, with Flood on hand to take the scoring pass and add the conversion.
Warwick converted from the touchline and Munster closed out the half with another try from Howlett, who used Johne Murphy as a foil to nip between two covering players and make the line.
This reflects the fact that few politicians have heard of Johne's, and the industry seems unsure how much to tell them about a disease which carries a stigma even in the cattle industry.
ECONOMIST: Another cow disease with a possible link to humans
The Tigers pack confirmed their dominance by earning a penalty try before driving Crane over for his second score and Johne Murphy completed his hat-trick in injury time as Leicester won in some style.
At one stage the Tigers had been languishing in seventh place when Quins were leading, but second-half tries from Johne Murphy and Varndell ensured Loffreda's side will face Premiership table-toppers Gloucester at Kingsholm on 18 May.
The match looked to have slipped away from the Premiership side when Johne Murphy was the first to miss in the shoot-out but James could not convert the opportunity and, after Martyn Williams also missed from the 22, Crane stepped up to be the match-winner.
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