Durham scored 153 for eight from their 20 overs and reduced Northants to nine for three before Cobb hit three sixes and 10 fours off 48 balls to put his side on the brink of victory.
Captain Alex Wakely also chipped in with a vital 43 off 39 deliveries and Northamptonshire reached their target with five balls to spare.
Northamptonshire, second bottom of county cricket's second division and beset by financial problems, also won the domestic Twenty20 title in 2013 and finished runners-up to Lancashire last season.
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In the semi-finals at Edgbaston earlier on Saturday, Northamptonshire beat Nottinghamshire by eight runs and England all-rounder Ben Stokes struck 56 off 36 deliveries to inspire Durham to a seven-run victory over Yorkshire.
(Reporting by Ed Osmond; Editing by Tony Jimenez)
