Rain cut the match short when England were 103 for three in 25.1 overs in reply to South Africa's 175 for nine in 50 overs.
England won by 26 runs on the Duckworth/Lewis method.
Vaughan was named man of the match but he said England's win was set up by their bowlers who kept South Africa under pressure from the start after he won the toss and sent the home side in to bat in overcast conditions.
"Our bowling was outstanding," he said.
South Africa hit back early in the England innings and the tourists were struggling at 44 for three before Vaughan and South Africa-born Kevin Pietersen (22 not out) first steadied the innings, then went on the attack shortly before the rain struck to make sure their side were comfortably ahead on the comparative tables.
South African captain Graeme Smith said poor batting had caused his team's downfall in the opening encounter of the seven-match series.
"It wasn't easy to bat up front but there were some loose shots," he said.
Looking ahead to the second match in Bloemfontein Wednesday, Smith added: "We have a lot of improving to do."
South Africa crashed to 90 for seven before the lower order batsmen, led by Shaun Pollock (37) and Nicky Boje (33) enabled them to make a partial recovery.
Opening bowlers Darren Gough (two for 27) and Matthew Hoggard (two for 35) did the initial damage but they were well backed up by seamer Kabir Ali (two for 29) and left-arm spinner Ashley Giles, who took three for 18.
Vaughan surprisingly used Giles in the last over of the innings and he responded by taking the wicket of Pollock.
South Africa made a disastrous start, with opening batsmen Smith and Herschelle Gibbs falling within three balls of each other inside the first three overs.
Smith chased a wide ball from Gough and was caught at second slip, while Gibbs was caught at cover, driving too early at Hoggard.
Adam Bacher, returning to the side after an absence of seven years, was bowled off an inside edge when he charged down the pitch against Hoggard after finding it difficult to get the score moving.
Jacques Kallis made only five off 26 balls before being caught down the legside by wicketkeeper Geraint Jones off Kabir Ali, who was bowling for the first time in a one-day international.
In his only previous appearance, against Zimbabwe in 2003, rain washed out the game before he had a chance to bat or bowl.
Ashwell Prince and Justin Kemp both topped 20 and Mark Boucher made 17 but South Africa were in deep trouble at 90 for seven when Boje joined Pollock in a stand of 58 off 81 balls. The stand was interrupted briefly by rain which drove the players off the field after 35 overs.
England were off to a brisk start but Makhaya Ntini struck in the fourth over when he had Trescothick caught at second slip when the batsman was trying to withdraw his bat.
Experimental opener Jones edged a Pollock outswinger to wicketkeeper Boucher in the next over.
The hopes of the majority in the capacity crowd of 31,500 were raised when Andre Nel's second ball brought the wicket of Andrew Strauss, the hero of England's Test series win before the one-day matches.
Strauss clipped a catch to square leg.
Former South African Kevin Pietersen had a hostile reception from the crowd and there was some tense cricket as he and Vaughan went four overs without scoring a run against Pollock and Nel.
But a change of bowling released the pressure. Kemp went for ten runs in his first over and Boje for eight. Another nine were scored off Kemp's next over with Vaughan in particular striking the ball cleanly.