Derbyshire's star was Martin Anderson, who took six wickets and also mustered a half century.
It meant disappointment for another Aussie, Ashton Turner, whose 53 from 22 balls was not enough to save Leicestershire.
Andersson produced one of the best bowling performances in the history of the competition as the Foxes were dismissed for 145 in 15.2 overs.
The all-rounder ran through the Leicestershire innings to finish with figures of 6-23 from his four overs.
Falcons had earlier made 5-230 with their top four batters all contributing, including 57 from Andersson.
Aneurin Donald struck 49 from 24 balls before falling to teenager Alex Green's first delivery with the score on 78.
Andersson and Jewell, who hit seven boundaries, both made half-centuries and Matthew Montgomery's 27 from 13 balls ensured the Foxes would face a testing target.
Leicestershire's reply was soon in disarray as they lost three wickets for six runs, two of them to Ben Aitchison.
A fourth-wicket stand of 92 between Ashton Turner and Ben Cox gave the Foxes hope, but Andersson then took over as the Falcons kick-started their North Group campaign in style after two previous defeats.
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