Worcester scored two early tries through Andy Symons and Jake Abbott but they failed to capitalise on their first-half superiority by missing five kicks at goal, which would have given them a handy lead.
Two penalties and a drop-goal from Owen Williams had kept the Tigers in contention in the first half, with the sin-binning of Sam Betty just before the interval being the turning point of the game.
In his absence, Michael Noone put Leicester ahead with their opening try before a penalty try award sealed the Warriors' fate - with Christian Scotland-Williamson's last-minute effort gaining Worcester a bonus point.
The hosts included James Percival for the first time this season following a six-month lay-off with a knee injury. Centre Symonds made his full debut while Mariano Galarza was released by Argentina to join Percival in the second row.
Leicester gave Matt Smith his first starting appearance of the season with Miles Benjamin picked on the wing to face his former club. There were also recalls for number eight Jordan Crane and scrum-half David Mele.
Worcester had the first chance for points but Symons was off target with his long-range penalty attempt - however, he soon atoned by finishing off a splendid movement for the opening try.
It began when Warriors counter-attacked from inside their own 22 to send Max Stelling away on a 40-metre run. Stelling then kicked ahead before combining cleverly with Symons to set up a ruck.
Worcester recycled allowing Galarza to burst away and present Symons with a simple run-in.
Owen Williams had Leicester's first opportunity but his penalty kick rebounded back off a post, before Worcester extended their lead.
Tigers wing Henry Purdy failed to get a hand on an excellent up and under from Paul Warwick, enabling Symons to seize on the loose ball and send Jake Abbott away for the try.
Warwick again missed with the conversion before a Williams penalty for the visitors made it 10-3 to the Warriors at the end of the first quarter.
Worcester lost Abbott through injury but they continued to be the better side. However, they could not take advantage as Symonds, who had resumed the kicking duties, missed badly with a routine penalty attempt.
Warriors were made to pay for their profligacy as Leicester had their best spell of the half.
Williams kicked another penalty before Benjamin powered through to pressurise the home try line.
Betty was penalised and yellow-carded for killing the ball before a Williams drop-goal with the last kick of the half reduced the arrears to only one point.
Symons departed at the interval through injury before Tigers took the lead for the first time from a close-range scrum, Crane's pass setting up Noone for an easy try.
Back came Worcester but Warwick, restored to the task of goal-kicking in the absence of Symons, again fired wide before there was some relief for the home supporters with a first success for the outside-half.
Leicester led narrowly 14-13 at the end of the third quarter but their pack was becoming the dominant force. Both Smith and Jamie Gibson were hauled down narrowly short by some desperate home defence but the pack did gain their reward when they were awarded a penalty try as the Warriors could not cope with a scrummage drive.
Gibson was yellow-carded to allow Warriors a sniff of victory. They did manage a try from replacement, Scotland-Williamson, a minute from time but Tigers held on.