Asked to chase 217 off 40 overs, Sarisbury got home by four wickets with eight balls remaining of a fascinating tie at their Allotment Road base.
Franklin, batting at No 6, provided the late-innings impetus with an undefeated 58 off 39 balls with five fours and three sixes.
Sarisbury were 110-4 when Franklin walked to the crease, where he helped skipper Ricky Rawlins (72) add 81 for the fifth wicket.
Rawlins and Franklin have both been Southern Premier Division title winners in the past and the pair cashed in as the Dons bowlers were punished for repeatedly dropping short of a length.
With Sam Floyd switching his leisure pursuits to golf, Sarisbury now rely heavily on Rawlins and Josh Hill, who eased the hosts into the driving seat at 87-1 after opener Eliott Smith (7) fell early.
Hill whacked a run-a-ball 40 in adding 78 for the second wicket with his skipper before miserly one-time Hampshire left-arm spinner Ian Turner (1-17 off eight overs) bowled him.
Franklin and wicket-keeper Simon Orr (12 not out) struck six boundaries in quick succession to seal Hambledon’s fate.
Earlier, Hambledon - who play two divisions lower than Sarisbury in the Southern Premier League structure - were reasonably satisfied with their batting.
The top six all made a start - captain Spencer Le Clerq’s 22 was the lowest score - with Henry Glanfield (41), Chris Pratt (37) - back from Loughborough University for the weekend - and Indy Chakribati (36) top scoring.
Teenager Max Martin and opening bowler Jordan Wright both claimed 2-33 - the latter’s were tailend wickets - as Hambledon closed on a competitive 216-8.
Sarisbury have form in this competition - two years ago they reached the national semi-finals, standing just one game from a Lord’s showpiece occasion. But they were beaten by Reed, who recovered from 104-5 to chase down 196 with no additional dismissals and with more than an over to spare.
Rawlins’ side now visit Calmore, while Bramshaw face Sparsholt in the other group 26 semi-final.
Bramshaw defeated Sway by one wicket at the weekend, chasing 171 for victory after Daniel Bailey’s 4-17 off eight overs had reduced them to 114-7.