Cobb led by example, clubbing seven fours and a six in his 52-ball innings, as the Foxes chased down a victory target of 159 with 14 balls to spare.
Bangladesh star Shakib Al Hasan struck a blistering 43 off 21 balls in a fourth-wicket stand of 67 with Cobb as the Outlaws lost for the first time in five games in the competition this season.
Elsewhere in the North group, Phil Mustard hammered Durham's highest individual Twenty20 score of 91 in a 76-run home victory against Yorkshire at Chester-le-Street.
The left-hander struck 11 fours and three sixes in scoring his runs off 52 balls as Durham amassed 215 for six, their joint second highest T20 total.
Without their England men and injured skipper Andrew Gale, Yorkshire quickly slipped to 22 for three in reply and finished on 139 for eight.
Jim Allenby produced a fine all-round performance to help unbeaten Glamorgan to their fourth win, thrashing Somerset by nine wickets at Cardiff.
In front of a crowd of nearly 9,000 Glamorgan, who remain top of the Midlands, Wales and West Group, restricted Somerset to 125 for six with a fine bowling and fielding performance.
And Allenby (47 not out from 47 balls), who also had bowled four tight overs, and Chris Cooke (42 not out from 28 balls) helped to knock off the 126 needed with 4.4 overs to spare.
Varun Chopra completed a match-winning double for the week as Warwickshire finally registered their first victory in this season's competition as they beat Worcestershire by seven wickets at New Road.
Captaining the side in the absence of the injured Jim Troughton, Chopra made 65 from 52 balls in an impressive follow-up to his double century in overcoming Middlesex in the LV= County Championship at Uxbridge.
Facing a Worcestershire total of 188 for five - built around Moeen Ali's 85 against his former county - Warwickshire got home with an over to spare as veteran Darren Maddy managed to outshine Chopra.
The 39-year-old all-rounder, who retires at the end of the season, turned the clock back to his early career with Leicestershire in racing to an unbeaten 84 from 42 balls.
Middlesex crushed Essex by eight wickets at Chelmsford after the home side were bowled out for 74, their lowest-ever total in the competition.
Only Owais Shah, with 39 from 43 balls, posed any threat as the rest of his colleagues were swept aside with embarrassing ease in this South group match.
Kyle Mills finished with three wickets for four runs and Middlesex had no difficulty in carving out a victory that enabled them to join Surrey level on points at the top of the table.
Neil McKenzie hit the last ball of the match for four to guide champions Hampshire to a four-wicket win over Sussex in the South group clash at the Ageas Bowl.
Experienced McKenzie kept his nerve as wickets tumbled in Hampshire's pursuit of Sussex's modest total of 118 for seven. They went into the last over from Scott Styris requiring six to win and a single was needed off the final ball.
As Sussex crowded the South African with a ring of close fielders, McKenzie lifted his shot to midwicket, thereby ensuring Hampshire's third win in four fixtures.
Tight bowling by spinners Simon Kerrigan and Arron Lilley aided by some athletic fielding helped Lancashire to defeat Derbyshire by 12 runs in the North group floodlit match at Old Trafford.
The young pair returned identical figures of one for 25 from their four overs as Steven Croft's side restricted the visitors to 153 for six in reply to Lightning's 165 for nine.
The win was Lancashire's third in six games and it boosts their hopes of qualifying for the quarter-finals.
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