Spearman hit the winning runs, by which time he had faced 66 balls and collected six boundaries.
Craig Spearman (51 off 28 balls) and Ian Harvey (28 off 17) gave the Gladiators a good start on Monday.
BBC: SPORT | Cricket | Counties | Gloucestershire make last eight
That was after a turbo-charged start by Spearman and Harvey who put on 74 in 36 balls at the start.
BBC: SPORT | Cricket | Counties | Gloucestershire make last eight
But Spearman, resuming his old role, led from the front as Gloucestershire reached their target with more than 12 overs to spare.
Spearman soon fell to Kenyan off-spinner Jamie Dalrymple with the score on 115 for four after hitting seven fours and three sixes.
The Gladiators set off well in pursuit, openers Craig Spearman (37) and Jack Russell (31) putting on 68 for the first wicket.
But this surge fell away when first Spearman, Stephen Fleming and then Craig McMillan were sent back to the pavillion with the scores at 71 for three.
But Spearman instantly fought back, Tim Hancock contributing just 22 to a third-wicket stand of 94 before becoming the first of Paul Weekes' two victims.
First Kadeer was bowled by a Jones leg-cutter, before Marshall was bowled playing no shot to Groenewald and Spearman edged a wide Jones delivery behind.
The study found there was a positive link between good information technology and national test results, of 0.07 under the Spearman Rank Correlation statistical measurement system.
Hodge (73) failed to get far enough forward to a good-length ball and edged to Russell before Pattison (27) became Spearman's second catch at first slip.
Gloucestershire still faced an uphill task and it became mountainous when Craig Spearman cut Lucas to point, Hodnett was caught behind and Kadeer Ali was trapped lbw.
BBC: SPORT | Cricket | Counties | Northants on course for third win
Tim Hancock (three) edged England paceman Harmison to 'keeper Andrew Pratt but a third-wicket stand of 103 between Spearman and Windows appeared to put Gloucestershire in control.
In reply, Spearman and James Pearson shared an opening stand of 38 before the latter pushed forward at Mark Davies and was caught at second slip by Hodge.
Earlier, Mike Smith trapped opener Ali Brown lbw with the fourth ball of the day and then saw Mark Ramprakash nick one to Spearman at first slip for 11.
After a slow start to the New Zealand innings, Nathan Astle and Craig Spearman looked to replicate the performances of the West Indies openers when they hit Franklyn Rose for 15 runs off his third over.
Meanwhile, in Albany and other towns in Georgia's peanut-growing regions, trailers and barns were overflowing with peanuts after the usual storage bins overflowed, said Tyron Spearman, executive director of the National Peanut Buying Points Association.
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