Resuming on 213-2 with Markram and captain Temba Bavuma unbeaten overnight, South Africa needed just 69 runs with eight wickets in hand to lift the mace at the 'home of cricket'.
They lost the injured Bavuma early on after he was caught behind off Pat Cummins, adding just one to his overnight tally to leave for 66.
It was him and Markram who had put the South Africans into such a good position on day three, and Markram continued to chip away at the total. He saw Tristan Stubbs fall for eight to give Australia a glimmer of hope, but the opener chipped away to put the game beyond doubt.
"We prepared hard, we came in with a lot of belief but with a lot of doubters," said captain Temba Bavuma.
"It's a special moment for us and people back home, probably it will sink in in a couple of days.
"The energy was there, I think us as a team have been wanting this. We've been relentless, getting to the doorstep consistently and experiencing heartache, the sun's been with us. Hopefully, this win is one of many," he added.
He eventually fell for with just seven needed to win, caught at square leg as he looked to finish the test in a hurry.
Congratulated by every Australian player, Markram was given a standing ovation by those at Lord's, with his job done.
Kyle Verreyne hit the winning run to start the celebrations and end 27 years of hurt on the international stage.
Nervy finish
"It was the most nervous I’ve ever been. I didn’t want to come out and bat but when Aiden went out I had to," said Verreynne.
South Africa started the fourth day favourites to win but nervy, given a litany of past disasters, and would have been further on edge when captain Bavuma was out in the third over of the morning.
Markram and Bedingham then diligently put together a 35-run partnership to see them to the brink of success, before the win was secured in the last over before lunch.
South Africa, who came into the contest as underdogs, won the toss on the opening day and gambled by putting Australia into bat. But with Kagiso Rabada returning figures of 5-51, they were vindicated as the defending champions were dismissed for 212.
South Africa’s reply of 138 meant Australia had a 74-run lead, significantly increased after scoring 207 in their second innings, with their tail order recovering after they had slumped to 73-7.
But South Africa profited from a dramatic change in the pitch conditions - after 24 wickets fell on the first two days – to patiently bat themselves to a landmark victory.
"We didn't bat out the opposition after a decent first-innings lead," said Cummins. "South Africa didn't give us a chance in the fourth innings."