Harmer claimed 4-21 as India, chasing 124 for victory, folded for 93 in a remarkable batting meltdown in front of nearly 40,000 crowd at Eden Gardens.
Earlier, Temba Bavuma dragged South Africa back into the contest with an innings of tremendous grit.
Having conceded a first innings lead of 30, South Africa resumed day three on a precarious 93-7 with the onus on Bavuma to give his bowlers some runs to defend.
Bavuma replied with a heroic unbeaten 55 - the only fifty by any batter on a minefield of a pitch where inconsistent bounce and spin made life difficult for the shot-makers.
Corbin Bosch produced a defiant 25 but once Jasprit Bumrah pegged back his off stump, South Africa's tail promptly caved in.
Bavuma was left stranded after a 136-ball defiance, which contained four boundaries and plenty of determination.
Chasing a small total to win, India were already short of a batter as skipper Shubman Gill was deemed unfit to bat in the second innings having to retire hurt on Saturday with a neck injury.
To make it worse, Marco Jansen claimed two wickets in his first seven deliveries to leave India reeling.
"He (Harmer) was phenomenal, he showed his class, he showed why he has 1000 FC wickets. It's a privilege to bowl alongside him," South Africa spinner Keshav Maharaj said.
"Take nothing away from (seamer) Marco Jansen as well. What a team effort, there has been exceptional team effort as well. Really great desire to get of the line, and we did it.
There was a ball with your name on it (as a batter). Simon showed the consistency more than me. Also the faith Temba showed in us, a lot of credit to him.
"We want to go forward and we can be really thrilled with what we have achieved."
Yashasvi Jaiswal fell for a four-ball duck while opening partner KL Rahul made one before he too perished caught behind.
Harmer dismissed Dhruv Jurel and Rishabh Pant to reduce India to 38-4 and then trapped Ravindra Jadeja (18).
Aiden Markram struck a crucial blow when he dismissed a set Washington Sundar (31) in the slip.
Realising the futility of trying to hang around, Axar Patel smacked a couple of sixes before falling for 26, which effectively sealed the match in South Africa's favour.
"After a game like this, you can't dwell into it too much," India vice-captain Rishabh Pant said. "We should have been able to chase it. The pressure kept on building. We didn't capitalise enough.
"Temba and Bosch had a good stand in the morning. That partnership between them hurt us. There was help from the wicket.
"A score of 120 can be tricky on these surfaces. But having said that, we should be able to soak in the pressure and capitalise. We haven't thought on the improvements but we will come back strong for sure."