The Warriors' points all came from the boot of Shane Drahm as he kicked five first-half penalties while Will Skinner scored the only try of the match.
The turning point came when Ben Kay was shown a yellow card for an off-the-ball challenge on Craig Gillies.
Worcester made the extra man count, went 15-8 ahead at the break and survived a late onslaught for the win.
Leicester head coach Pat Howard:
"We didn't play anywhere near where we could have. "There are no excuses. We had opportunities to win and we didn't take them.
"We lost two guys in the sin-bin during the match and those were difficult periods for us."
Worcester: Le Roux; Havili, Hinshelwood, Lombard, Delport;
Drahm, Gomarsall; Windo, C Fortey, Tuamoepeau; Murphy,
Gillies; Horstmann, Tuamoheloa, Hickey.
Replacements: Vaili for Tuamoheloa (73).
Not used: McDonald, L Fortey, Blaze, Powell, Brown, Tucker.
Leicester: Vesty; A Tuilagi, Lloyd, Hipkiss, Varndell;
Broadfoot, Healey; Holford, Chuter, Moreno; Hamilton, Kay;
B Deacon, Skinner, Johnson.
Replacements: Smith for Lloyd (66), Goode for Broadfoot (51), Morris for Holford (66), Cullen for Hamilton (61).
Not Used: Buckland, Croft, Cole.
Sin Bin: Kay (32), Hamilton (47).
Att: 9,726
Referee: Roy Maybank (Kent).