It was Gloucestershire's ninth win in 10 Lord's one-day finals - and the perfect way for Geraint Jones, who had earlier hit a half-century himself, to bow out in his final game before retirement.
For dejected opposite number Gareth Batty - caught by man-of-the-match Jack Taylor in the final over to spark wild scenes of Gloucestershire jubilation - it was the former Worcestershire and England spinner's third Lord's final defeat, all of them by Gloucestershire.
They did not appear to have put enough runs on the board despite a Jones-inspired recovery from 108-5 to 220-7, in which Taylor contributed 35 off 26 balls.
And 15 deliveries were left unused after Dernbach finished off the innings by claiming the wickets of Jones (50), and then Craig Miles and David Payne for successive first-ball ducks.
Surrey lost openers Jason Roy and Steven Davies early during a superb new-ball spell by James Fuller (2-34), but Sangakkara and Burns gradually got on top.
However, just when Surrey looked to have the game in their grasp, both were out in the space of 13 balls and the tail buckled under mounting pressure as they suffered their sixth defeat in 11 Lord's finals.
It briefly looked as if 17-year-old Sam Curran (37) might steer Surrey home against the county for whom his late father Kevin used to play.
But he holed out on the long-on boundary off the first ball of the final over, and skipper Batty followed two balls later, picking out Taylor at deep mid-wicket.
Following the injury that ruled out Zafar Ansari, Surrey called 40-year-old former Pakistan all-rounder Azhar Mahmood into their side, two days after his appearance at The Oval in a Cricket for Heroes match, in which he dismissed Brian Lara and Brendon McCullum.
This time his 10 economical overs brought him 2-28, but his departure, stumped for five, was one of a series of rushes of blood which cost Surrey the chance of silverware.
Team-mate Sangakkara's innings may have been his last at Lord's now he has retired from international cricket.
It did not match the stunning 166 he scored in the semi-final win over Nottinghamshire, but he played faultlessly, getting the majority of his runs in ones and twos and hitting just three boundaries, until uncharacteristically gifting his wicket to spinner Jack Taylor.
Skipper Batty was one of only four Surrey players who played in their last Lord's winning team in 2011, but this time ended up on the losing side despite winning the toss and seeing his decision to bowl first apparently pay off.
If they were disappointed, ex-England wicketkeeper Jones was elated by Gloucestershire's victory, having played a sensible last innings to get them back in the game.
After reaching his 50 with a pull for four off Dernbach, he was bowled with the very next ball, but he had done enough to warrant a warm reception on his way back to the pavilion.