Heavy rain in the morning saw the match reduced to 34 overs per side.
Shiv Thakor took 4-49 for the hosts but a 104-run second-wicket partnership between Daryl Mitchell and Tom Fell (73) helped the visitors post 202-9.
In response, Greg Smith (34 not out) saw Leicestershire reach 93-2 off 15.4 overs before rain ended the match.
After Leicestershire won the toss and chose to bowl, both sets of players were forced to leave the field before the first ball was bowled as an unexpected burst of heavy rain interrupted proceedings.
When the players finally made it back out on to the field an impressive knock of 73 off 61 balls from Fell, which included 12 fours, held Worcestershire's innings together as Thakor, Charlie Shreck (2-26) and Ollie Freckingham (2-38) stifled the rest of the batsmen.
Although Charles Morris (1-19) and Joe Leach (1-25) reduced the home side to 45-2 early on in their reply, five fours from Smith ensured Leicestershire were well ahead of the D/L par score before rain intervened again.
Victory for the hosts means they can still qualify for the quarter-finals with just one spot in Group A remaining to be decided.
Leicestershire moved into fourth place on seven points and are away to Lancashire in their final match on Thursday, while Derbyshire sit in fifth place with six points and entertain third-placed Gloucestershire at the same time.
Leicestershire completed a 16-run One-Day Cup victory over Worcestershire under the Duckworth/Lewis method to keep alive their quarter-final hopes.