After a poor opening quarter, Leicester took the lead through Julien Dupuy's 22nd-minute penalty.
Worcester should have levelled soon after but fly-half Joey Carlisle missed with a kick at goal and Leicester took advantage.
Argentine prop Ayerza scored the first try after rounding off a move which was started by Aaron Mauger's misplaced pass, while Tuilagi charged through some flimsy tackling to run in from halfway.
Leicester lost momentum at the start of the second half and Bowley finished a sweeping move to give Worcester hope.
But the visitors had captain Pat Sanderson and replacement prop Aleki Lutui sin-binned and a burst of three Tigers tries in eight minutes settled the contest.
Replacement scrum-half Youngs crossed from close range, Tom Croft then set up Vesty and centre Matt Smith powered over unopposed.
Leicester acting head coach Richard Cockerill: "It's no secret that we are trying to play with the ball in the middle a bit more.
"I thought the endeavour was great but the execution was a bit poor and scrappy at times.
"It's hard against a team that don't want to play too much and they make you fight for every bit of contact.
"But I'm obviously happy with the result and we need to start putting some pressure on the teams above us.
"We have not been used to playing like that and some of it is because Worcester stifled and frustrated us for long periods and we tried to force it a bit.
"We scored some good tries, we bombed some other great opportunities. But, all in all, you have got to take the result.
"It was a good win for us and something to build on but we will have to be sharper for London Irish next week."
Worcester director of rugby Mike Ruddock:
"We have got Wasps and Northampton at home and Bristol away - those are three massive games for us, our cup finals.
"We had to make a lot of tackles last week and the game before. To come to Leicester and get a result is going to take an effort from all 22 so I felt it was appropriate to share the load today.
"Obviously we feel we are not playing to our full potential but next week we will hopefully get Chris Latham and Hal Luscombe back - and they will make a big difference.
"It would have been nice to have got a win at Leicester but it wasn't really a target game for us. The next three are massive target games for us."
Leicester: G Murphy; Hamilton, Smith, Mauger, Tuilagi; Vesty, Dupuy; Ayerza, Kayser, Castrogiovanni, L Deacon, Kay, Newby, Woods, Crane.
Replacements: J Murphy for G Murphy (52), Erinle for Tuilagi (66), Youngs for Dupuy (58), Chuter for Kayser (63), White for Castrogiovanni (48), Croft for L Deacon (64), B Deacon for Crane (55).
Sin Bin: White (77).
Worcester: Walker; Gear, Rasmussen, Crichton, Garvey; Carlisle, M Powell; Morris, Fortey, Tuamoepeau, Bowley, Gillies, T Wood, Sanderson, Horstmann.
Replacements: Grove for Crichton (39), Benjamin for Carlisle (70), R Powell for M Powell (78), Mullan for Fortey (49), Lutui for Tuamoepeau (59).
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