Set an improbable target of 331 after England had lost their nine remaining second-innings wickets to be dismissed for 233, the touring side never came to terms with a spiteful pitch of uneven bounce and were bowled out for 119.
The win was England’s first against South Africa at Lord’s since 1960.
Root simply had to rotate his bowlers, although spinner Moeen Ali took the honours with figures of 6-53, giving him 10 wickets in a man-of-the-match performance that included an invaluable first-innings knock of 87.
“It was definitely the best all-round performance of my career,” Moeen said. “The pitch allowed me to attack their batsmen as much as I could.”
Moeen produced turn and bounce on a wicket that seldom favours fourth-innings run chases. Only three teams have won a test match by scoring more than 300 at the home of cricket, but the tourists must have felt they were in with a chance after dismissing England so cheaply by mid-afternoon.
The size of their task, however, became clear when Heino Kuhn (9) and captain Dean Elgar (1) both fell with the score on 12 to fine catches from keeper Jonny Bairstow and Moeen off his own bowling.
JP Duminy also departed cheaply and once Hashim Amla was trapped lbw by Liam Dawson, it was only a matter of time.
England had their share of luck when Quinton de Kock chopped the ball on to the stumps off his foot but they thoroughly deserved the win against a side who made too many mistakes in the field.
The final wicket fell when Morne Morkel heaved Dawson to Keaton Jennings on the boundary.
Earlier, England’s total was boosted by combative batting by Bairstow, who was dropped off a simple chance to Vernon Philander on seven before making 51, and tailender Mark Wood.
That pushed the lead past 300 as South Africa’s bowlers struggled to match their morning success when England, who resumed on 119-1, lost seven wickets.
Morkel claimed three victims and Keshav Maharaj four as England failed to force the pace. Alastair Cook top-scored with 69.
Root said he was delighted by his first victory in charge, which he owed much to the runs scored by England’s top three in difficult circumstances on Saturday night.
“It is nice to stand here having made all the right choices,” he said with a smile. “But the whole team played a big part in this win. Moeen made it very difficult for the batsmen to take him on.”
The second test in the four-match series starts on Friday in Nottingham with South Africa captain Faf du Plessis set to return. Fast bowler Kagiso Rabada, however, is suspended.
(Reporting by Neil Robinson, editing by Ed Osmond)