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The visitors made a strong start to the match as Vereniki Goneva scored two tries in the opening 25 minutes, with his first coming from a solo effort just three minutes after kickoff.
However, Leicester gave away a string of penalties in the first half, allowing Chris Pennell to keep the home side in contention with four kicks in just over 10 minutes.
Worcester edged ahead for the first time early in the second half when Josh Drauniniu went over in the corner.
However, Toby Flood was able to restore the Tigers' advantage inside the final 10 minutes with his third penalty of the match, sealing the victory.