The 26-year-old all-rounder hit a one-day best of 73 to give Gloucestershire's innings a look of some respectability after they had been in early trouble at 58 for 5. Then it was Hancock's smart fielding on the extra cover boundary that gave Gloucestershire a one-run victory in a thrilling finish.
The England Under-19 player Paul Franks needed to hit the last ball for four to win the match for Nottinghamshire. But Hancock thwarted that effort and also stopped Franks from pinching a third run that would have earned the home side a tie.
Mark Alleyne must have been questioning his decision to bat first when his side soon hit trouble, as Kevin Evans claimed two early victims to set the tone for the first half of Gloucestershire's innings. But Hancock joined Jack Russell at the crease and proceeded to share a sixth-wicket partnership of 98 in 18 overs.
Russell was eventually dismissed by Franks for a workmanlike 40, but Hancock went on to reach 73. That performance was chiefly responsible for Gloucestershire rattling up 93 runs off the last 10 overs.
Nottinghamshire made a poor start to their run chase and lost wickets at crucial points in the early stages, and it was only a half century from Noel Gie, who hit 56, which kept them within range.
There was another close finish at Uxbridge, where Essex fell short on 198 for 7 to lose by two runs against Middlesex.
The Essex opener Darren Robinson made 83 but was then run out and, despite Stephen Peters making 54, it was not quite enough against their Benson and Hedges Cup quarter-final opponents. Middlesex had earlier made 200 for 8, with Mark Ramprakash top scoring with 57.
Leicestershire also finished just short in their run chase at Derbyshire despite the best efforts of Paul Nixon, who ended up on 50 not out in their seven-run defeat.
Derbyshire had set what looked a highly attainable target after being restricted to 190 for 6, in which their Australian opener Michael Slater was the top scorer with 68. But the home side's bowlers put the brakes on Leicestershire's scoring, and Nixon's unbeaten half-century was their largest contribution as they were restricted to 183 for 7.
Tony Penberthy's unbeaten 79 steered Northamptonshire to a seven-wicket victory over Somerset at Taunton.Reuse content