Gloucestershire were able to build on a prolific initial six-over powerplay, in which they posted 67-1. Dawid Malan impressed with 28 off 19 balls in an opening stand of 46 in 4.4 overs with Short, who took 17 off the sixth over, sent down by Jordan, including two sixes.
Miles Hammond and Liam Scott fell cheaply to Dan Lawrence and Tom Lawes respectively, but Taylor hit his first ball, from Lawes, for four and added a six and two more boundaries in facing 26 balls.
Short went to a 33-ball half-century, taking advantage of some inconsistent Surrey bowling on a pitch with more pace than the usual Bristol surfaces.
The visitors were forced to bring an extra fielder into the ring in the 19th over having failed to meet the cut-off time. Kamran Dharilwal had just been caught off a Sean Abbott no-ball and sent the resulting free hit straight for six.
It proved his only scoring shot, but Short, who had offered one difficult caught and bowled chance to Jordan on 49, otherwise maintained concentration to the end of the innings to ensure his side of a competitive total.
Surrey’s reply got off to a bad start when Laurie Evans was caught behind off Scott in the second over attempting a ramp shot. Jason Roy quickly responded with a straight six off Matt Taylor, but fell to the next delivery, miscuing a shot to deep square to leave his team 14-2.
Hayes went for 16 off his first over, but Marchant de Lange quickly removed Josh Phillippe and at the end of the powerplay, Surrey were 48-3.
The unflappable Hayes was visibly thrilled to take his first wicket for his new county, the prize one of Ollie Pope, who had made 20 when top-edging an attempted leg-side hit to be caught at cover and Surrey were in deep trouble at 53-4 in the eighth over.
Tom Curran was yorked by left-arm spinner Graeme van Buuren before Hayes struck again, Lawes caught and bowled off another top-edge to make it 59-6.
Abbott was stumped to the bowling of Van Buuren and at 62-7 the home side had a complete stranglehold. Despite some belligerent blows from Lawrence and Jordan, they never looked like recovering and when Lawrence fell to Scott with the total on 110 any doubt about the outcome was erased.
Hayes claimed his third wicket in the final over and Gloucestershire could reflect on a superb team performance.
The result leaves both sides second in their respective groups.
Match report supplied by ECB Reporters’ Network, supported by Rothesay.