Leicestershire, chasing a victory target of 288, looked in trouble at 128 for 4 in the 26th over, but the 23-year-old Dakin and Darren Maddy, 21, shared a fifth-wicket stand of 153 in 23 overs to transform the match.
Both batsmen compiled career-best scores in the competition - as did Durham's John Morris earlier in the day. His 145 was not only a personal best but also Durham's highest individual innings in any form of one-day cricket.
Gloucestershire secured a place in the quarter-finals thanks to a 21- run win over Hampshire at Bristol. Chasing Gloucestershire's 272 for 9, Hampshire reached 162 for 3 but collapsed to 251 all out. Mark Alleyne won the gold award with 75 from 53 balls for the home side.
At Derby, Dean Jones guided Derbyshire to a six-wicket win over the Minor Counties. The Australian veteran scored 142 from only 101 balls.Reuse content