With a game in hand, they are now only five points behind their visitors, who had also won their first two matches.
Borthwick, who was 24 last week, played like a seasoned pro as he followed knocks of 94 and 51 not out at Taunton by making light of what threatened to be a tricky task when Durham resumed with 147 needed and eight wickets standing.
The difference proved to be the greater depth of the Durham attack as Ajmal Shahzad ploughed a lone furrow, taking all four wickets to go with his four in the first innings and take his season’s tally to 22 in three games.
Oliver Robinson could hardly be expected to sustain his dream debut, and although he beat Borthwick three times in one over the left-hander responded by hitting him for three fours in his next.
When Michael Richardson lost his leg stump to Shahzad in the day’s fourth over the contest was still in the balance.
Calum MacLeod, originally signed as a T20 specialist last year, took 25 balls to get off the mark.
He succeeded in seeing off the dangerous Shahzad and opened his account by driving Matt Hobden to the cover boundary.
On seven he edged Hobden past Ed Joyce’s left ear at first slip and went on to make 26 before attempting an upper cut, which flew straight to third man.
Paul Collingwood joined Borthwick for the final ten minutes before lunch and made light of knocking off the remaining runs, scoring 35 not out off 44 balls.
Borthwick hit two fours off the first two balls of a Robinson over to complete the task, leaving Robinson with none for 76 in 14.2 overs.
Mark Robinson, Sussex’s cricket manager, said: “ It was a good game on a bowler-friendly pitch, which made it exciting. There were some indentations, so the heavy roller became a tactical thing and was possibly their final trump card.
“They also had Chris Rushworth, who was the best bowler in the game for consistently hitting the right length. His natural length is the right length for this pitch.
“We had two great opportunities to get in a winning position, but John Hastings took the first one away from us with the bat. Then Rushworth made sure we fell a bit short in our second innings after a good start.
“Oliver Robinson had a dream debut, Matt Machan has continued his good form and Ajmal Shahzad has hit the ground running. He’s not cocky like some people think – he’s enthusiastic.
“We fought hard and came up a bit short, but we are missing a few bodies and I’m pleased with what we have done here. It’s a tricky place to come.”
Durham’s Scott Borthwick said: “The captain asked me if I wanted to get my century when I was on 89 and we needed 11, but I said there was no point in messing around. I’m just happy to get a good win.
“It was the sort of pitch where you never really felt in. We knew we didn’t need to score quickly but felt if we could get through the first hour it would get a bit easier.
“I was also there at the end when we won at Taunton. It’s a good feeling – you feel like you’ve earned a beer.”