There was also a typically calm and responsible 43-ball 46 from their captain Michael Klinger as Gloucestershire chased down Kent 160 for 5, in which Joe Denly played an almost lone hand with a superb 99 not out from 63 balls, with five sixes and seven fours.
Benny Howell, too, played his part by following up his 2 for 29 with 20 not out from 13 balls, including the winning four struck over mid off from the second ball of the final over, bowled by Adam Milne, with the scores level.
Left-hander Hammond had scored 51, with three sixes and five fours, when he hit the 31st ball he faced, the first of Calum Haggett’s second over, to wide deep mid wicket to end an opening stand of 76 with Klinger.
Hammond uppercut Haggett for a remarkable six over cover point in the third over, and then also clubbed leg spinner Denly and seamer Ivan Thomas for maximums.
Klinger, content to play a supporting role, nevertheless seized on a high full toss from left-arm spinner Imran Qayyum at the start of the 11th over to smash a six over long leg. The over, which also contained five wides when Qayyum speared a ball down the legside, ended up costing 18 runs and left Gloucestershire requiring just another 57 from the last nine overs.
To their credit, Kent’s bowlers made the visitors work hard for the win, with Klinger departing in the 16th over when he tried to reverse-hit Qayyum but could only pick out Denly at backward point.
Cockbain and Howell held their nerve, however, and the result was never in doubt once they had plundered 18 from the 18th over, bowled by Marcus Stoinis until he went off injured after bowling five balls, leaving Qayyum to finish it.
A six by Cockbain over mid wicket was followed by Howell driving a powerful four and then taking another four, through backward point, from Qayyum’s final ball.
Kent reached 50 without loss in the six-over powerplay, with Denly the initial aggressor with square cut and lofted offside offside fours in the third over, bowled by Matt Taylor, and a six swung over wide mid wicket against David Payne.
Daniel Bell-Drummond, starved of much of the early strike, got going in the sixth over by pulling Taylor for six and then lofting the next ball over mid off for four. On 19, however, he was perhaps unfortunate that a thigh-high full toss from Benny Howell was not called as a no ball as he mis-hit to deep mid-wicket.
At halfway Kent were 84 for one, with Denly hoisting Howell for six in the medium-pacer’s second over, and Heino Kuhn contributed a brisk 18 before being bowled sweeping at Graeme van Buuren’s left-arm spin in the 11th over.
Successive sixes by Denly off van Buuren in the 13th over, the first straight and the second over the head of the long on fielder, provided Kent with some much-needed acceleration but Sam Billings found it hard to get going and it was no surprise when, on 7, he hit his 13th ball straight to long on to give Howell his 100th T20 wicket.
Stoinis offered one meaty blow, a lofted four off his fellow Australian Andrew Tye, before hitting the same bowler to long off where Howell ran to his left to hold a good catch
Denly finished that same over, the 17th, by driving Tye wide of Howell for six but at the start of the next over Alex Blake was leg-before for a first ball duck trying to work David Payne to mid wicket and the only boundary conceded by Payne and Tye in the last two overs came from the final ball of the innings, which Denly crashed through square cover to end up a frustrating one run short of a fifth T20 century.
After Gloucestershire’s eight-wicket win, opener Hammond said: “We had to fight quite hard but I always thought 161 was gettable in the conditions and despite it being a used pitch. It’s good that we managed to bounce back after losing to Glamorgan on Friday evening.
“The bowlers did really well today as a group and they executed their plans throughout the Kent innings. Joe Denly played a great knock but all our bowlers did well to keep them to that score. David Payne has great skills with the white ball and he nailed them down today.”
Kent vice-captain Joe Denly said: “It was quite a tough pitch to bat on, but we would have liked at least 170. But I didn’t think we were quite at our best with the ball today, and I didn’t really set a good example with my own first over. We didn’t build up enough pressure on them, and if we had managed to do that in the middle overs then it might have brought more wickets and that is what brings the rate down.
“It’s important not to get too down, because we have been very good with the ball in the competition so far and we have defended totals very well. But we just weren’t quite at our best. Marcus Stoinis just felt a little bit of a hamstring twinge which is why he went off towards the end of that 18th over, but we don’t think it is that serious and we have a bit of a break now before our next game next Friday.”