Borthwick first polished off Surrey's second-innings with career-best figures of six for 70 before hitting 42 and featuring in a match-clinching second wicket stand of exactly 100 with opener Mark Stoneman as Durham successfully chased down a 181-run target.
Stoneman made a fine 67 to add to his first innings 77 and Durham's 21-point win is their third in five championship matches this season.
Stoneman's vital 26-over alliance with Borthwick was also a Durham second-wicket record against Surrey.
A Durham mini-collapse, from 107 for one to 126 for four, gave Surrey a last chance of forcing a win themselves, towards the end of the hard-fought and fascinating contest, but Ben Stokes was equal to the pressure with an unbeaten 35.
The final day began with Surrey just 115 runs ahead at 164 for six in their second innings and initially Gary Wilson and Gareth Batty flourished with the bat as they took their seventh wicket partnership to 46.
Two wickets in three balls, however, tilted the match Durham's way again, with 19-year-old off-spinner Ryan Buckley first sending back Batty lbw for 27 and then Borthwick striking with the first ball of the next over to have Wilson, on 26, edging a drive to Collingwood at slip.
From 201 for eight there was a little more resistance from the Surrey tail, but Borthwick picked up both Stuart Meaker for nine and Tim Linley for 19 - both caught and bowled - to end Surrey's second innings on 229 and leave Durham 73 overs in which to score 181 for victory.
Linley removed Will Smith for three at the end of the opening over of Durham's second innings, the batsman limply hanging out his bat to nick a catch to wicketkeeper Steven Davies.
But either side of lunch Stoneman and Borthwick steadied Durham nerves against the twin spin threat of veterans Batty and Gary Keedy.
Stoneman pulled Linley for six and went to his second 50 of the game in the next over with a four off Batty, who trapped Borthwick in front four balls later.
Stoneman batted for 107 balls, hitting five fours besides his six, but his dismissal - clean bowled by the last ball of Meaker's first over when the fast bowler was brought on for the 33rd over of the innings - was part of a small Durham wobble, with Dale Benkenstein lifting a catch off Linley to mid-off two balls later.
Stokes, however, saw Durham safely home and finished the match in style with successive fours off Stuart Meaker after Paul Collingwood was bowled by Keedy just 25 runs short of the victory.