Surrey captain Burns was joined in a decisive third wicket partnership of 64 in 14 overs by Jamie Smith, who scored 33 from 51 balls as a fourth innings target of 156 was chased down in 44.5 overs on a turning pitch.
There was a late twist to a well-contested match, however, when 16-year-old debutant James Coles – Sussex’s youngest first-class cricketer – dismissed both Smith and Burns in the space of four balls to leave Surrey 120 for 4.
Ben Foakes, on 1, then edged Delray Rawlins just short of slip but he survived to finish on 13 not out, hitting the winning blow when he pulled the same bowler for six after a necessarily circumspect fifth wicket stand of 37 in 18 overs with Will Jacks, who ended up unbeaten on 22.
Victory was eventually completed a little over an hour after lunch on a day which began with Sussex resuming their second innings on 109 for 9 after a torrid final session the previous evening against Surrey spinners Dan Moriarty and Amar Virdi.
They made 128 in the end, thanks to Stuart Meaker’s defiant 42, but a policy of controlled and sensible aggression by Surrey’s top order paid dividends as Burns and company never allowed Sussex’s own trio of young spinners to settle.
Jack Carson, the highly-promising 19-year-old off spinner, did remove Scott Borthwick for 11 in the eighth over, having been given the new ball, and Rawlins’ left-arm spin then accounted for Hashim Amla for the second time in the match, caught low at short leg, before the Burns-Smith alliance looked like taking Surrey to a comfortable win.
But Coles’ double strike when brought back to bowl his fourth over, first bowling Smith middle stump with an arm ball – his third ball back – and then having Burns stumped by Ben Brown after beating his forward push on the outside, at least concentrated Surrey minds again and the youngster finished with figures of 2 for 32 from 11 overs.
All-rounder Coles, a student at Oxford’s Magdalen College and brought up in the Oxfordshire age group teams before joining the Sussex Academy at the end of last year, is believed to be the second youngest player – after Pakistan’s Shoaib Malik – to take three wickets or more on first-class debut.
The Surrey chase began with Borthwick managing one straight six off Carson before he was caught at slip from one that turned and bounced, and Amla’s 18 included two sixes – straight off Carson and then pulled over the mid wicket ropes off Rawlins.
Surrey had earlier finished off Sussex’s second innings for 128, with off spinner Virdi taking the final wicket for figures of 4 for 40. Slow left-arm spinner Moriarty, the chief destroyer of Sussex’s second innings, ended with 6 for 70.
Sussex added another 19 runs before Meaker went down the pitch to Virdi but could only pick out Jamie Overton at long on after putting on 26 for the last wicket with No 11 Henry Crocombe, who remained 9 not out and hit one lovely on-driven four off Moriarty.