Notts were unable to build on the platform laid by their openers as a two-and-a-half hour rain delay brought about a truncated chase for the visitors, with Leicestershire prevailing by 15 runs on DLS.
Freddie McCann and Ben Slater scored 50/0 off the opening 10.1 overs of the Outlaws' innings, with 370 the initial target. After the restart, Notts required 55 more from 23 balls, and were ultimately able to muster 39.
Earlier, last year’s Metro Bank One Day Cup winners made 60/0 off their opening ten overs, despite Luke Fletcher beating the bat on numerous occasions.
The pair gave away very few chances after that, until Liam Patterson-White finally made the breakthrough when he removed Swindells for 31 after a fine catch in the deep by List A debutant Freddie McCann.
The Foxes were 164/1 at the halfway stage, before debutant Farhan Ahmed broke the 79-run stand with his first List A wicket, Freddie McCann once again taking a superb catch running in from the boundary's edge.
Hill and India international Ajinkya Rahane remained on the attack, however, with the captain dismissed for 81 when he bottom-edged Lyndon James’ delivery onto the stumps.
Peter Handscomb made a quick 29 (21) before Rob Lord removed the Australian, Patterson-White making the fine diving stop at mid-off. James picked up his third of the day in the final over, removing Ben Cox for 29.
Slater and McCann, the latter fresh from a strong showing on T20 debut at Headingley, made a composed start to the innings, punishing loose deliveries and rotating the strike to reach 50/0 when the showers arrived.
With a severely truncated chase awaiting, the Outlaws called upon the T20 experience of Tom Moores and Jack Haynes, with both openers retired out.
But with a requirement to attack from the outset, Haynes was pinned first ball, before James holed out from the next delivery.
Patterson-White and Montgomery also perished as the required rate continued to increase, but Moores was able to enjoy a productive final over, taking 16 runs from Ben Mike's final five deliveries.
The wicketkeeper-batter finished unbeaten on 23 from 11 balls, but the hosts prevailed by 15 runs.
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