Sussex 149 for 5 (Alsop 51*, Burgess 48) beat Surrey 148 (Narine 29, Bopara 3-18) by five wickets
After bowling out the hosts for 148 on a used pitch, Sussex looked out of it when they struggled to 57 for four after Sunil Narine, who took two for 18 from his four overs, produced a wicket maiden in the 11th over.
But then Burgess joined Alsop in a match-winning partnership of 77 in eight overs that shocked Surrey, who had won their first two Blast fixtures of the season, and propelled the Sharks to the brink of an opening South Group win.
Burgess, having just deposited Dan Worrall for a huge six into almost the second tier of the JM Finn Stand, was brilliantly caught at short third man from the last ball of the 18th over - slashing a low full toss to Nathan Barnwell, the substitute fielder, who hung on gallantly.
That left Sussex needing 16 from the last two overs, and then ten from the last after Sam Curran bowled a tight penultimate over, and Alsop was equal to the task of taking ten off Tom Lawes.
Left-hander Alsop hit the second ball through extra cover for three and, with James Coles taking a single from the next, levelled the scores by hammering Lawes' fourth ball back past the bowler for a lofted boundary.
Alsop then pulled Lawes over a ring field for the victory-clinching single and his 42-ball anchor knock included legside sixes off Cameron Steel and Worrall and four fours besides.
But it had been Burgess, on-loan from Warwickshire and originally a product of Surrey's youth system before joining Sussex for three seasons from 2017, who played the innings of the game. He took the lion's share of the 19 runs that came from the 15th over, off Worrall, including a remarkable square cut six that landed on the boundary rope.
That left Sussex needing 52 from the final five overs and Burgess, in all, hit two sixes and five fours as the Sharks shook off injury setbacks that included opener Ali Orr having to bat with a runner - but still hitting 24 from 15 balls - after hurting his knee trying to take a boundary catch early in the match.
Narine earlier hit three sixes in a quickfire 29 but he then became the first of Ravi Bopara's three victims in a canny spell of three for 18 from the 12th, 14th and 16th overs, and Surrey's innings rather fizzled out as their last four wickets tumbled for just two runs in nine balls.
Surrey at first took the attack to a Sussex bowling unit shorn of both their overseas players, Pakistan's Shadab Khan and Australia's Nathan McAndrew, who were rested as a precaution following their unfortunate on-field collision in the Sharks' opening game defeat against Somerset at Hove.
Laurie Evans took two legside fours off George Garton in the second over and then lifted Coles' left-arm spin for a six into the back tier of the Bedser Stand in a third over that brought 16 runs as Will Jacks also plundered two fours.
But Tymal Mills, introduced for the fifth over, made the breakthrough when Evans, making room to carve, lost his off stump on 23 and Jacks fell for 13 - splicing Henry Crocombe's fast-medium to mid wicket - as Surrey ended the six-over powerplay on 42 for two.
A quite brilliant low diving catch at backward point by Coles cut Sam Curran's innings short on just two, the fielder scooping the ball up right-handed as the England all-rounder slashed Archie Lenham's leg spin hard for what he must have thought would be a boundary.
Narine's mid-innings aggression helped to take Surrey to 82 for three by the halfway point as he and Tom Curran added 53 for the fourth wicket but then Bopara began to produce his own particular brand of medium-paced magic as Narine lofted to long on and Curran (23) skied to extra cover, where Alsop took a fine running catch over his shoulder.
Jamie Smith did bludgeon Lenham for two straight fours but on 19 was bowled by Bopara as he tried to sweep a straight ball after moving outside his off stump.
Mills returned to help his captain turn the screw on Surrey's lower order and Jamie Overton was bowled for 13 by a thunderbolt that hit his off stump and both Steel and Lawes were run out trying to come back for a second run - Lawes being dismissed without facing a ball.
The innings ended with three balls unused, Jordan Clark swinging and missing at Garton to go for just eight and leaving Surrey short of a par total and guilty of losing their way against some excellent Sussex out-cricket.
Tom Clark was Sussex's first wicket to fall, caught at extra cover off Worrall for 9 and after Orr was taken at backward point off Lawes the Sharks reply stuttered further when Bopara edged Narine's third ball behind and the West Indian mystery spinner also had Garton caught by keeper Smith for 7.
Alsop and Burgess, though, held their nerve in front of a big crowd and, with Narine bowled out at the end of the 14th over, an equation of 71 from six was achieved. Their stand, meanwhile, was a fifth wicket T20 record for Sussex against Surrey.