In a low-scoring match on a turning and inconsistent pitch, the hosts faltered in their chase to be bowled out for 93 in the second session on day three at Eden Gardens, giving the Test world champions an unexpected 30-run win.
South Africa skipper Temba Bavuma made the only half-century of the match with a defiant 55 not out in his side's second innings 153.
When India chased, only Washington Sundar, with a top score of 31 from 92 balls, showed any great application with South African off-spinner Simon Harmer returning figures of 4-21.
Gambhir, however, preferred not to blame the pitch but put defeat squarely on the shoulders of the Indian batters.
"There were no demons in this wicket," Gambhir told reporters.
"It was not an unplayable wicket. It was a wicket where your technique can be judged, your mental toughness can be challenged, and more important than that is your temperament.
"The point is that you should be able to know how to play turn. And this is what we asked for and this is what we got.
"I still believe that irrespective of how the wicket was, 124 was chaseable."
Opener KL Rahul made 39 in India's 189 in their first innings but South African bowlers, led by Harmer who took eight wickets in the match, limited the opposition lead to just 30.
In the chase, India were a batter short after skipper Shubman Gill was forced out due to a neck injury he suffered on Saturday while batting and is "under observation" in hospital.
"There are lot of youngsters in that dressing room but in Test cricket, more than skill, you need mental toughness," said Gambhir.
The 44-year-old Gambhir, who has lost nine of his 18 Tests as India head coach, has picked up a reputation for tough-talking and confrontation.
Earlier this year, on the tour of England, he was involved in an angry exchange with Surrey's head groundsman at the Oval after the curator told him to stay at a certain distance while inspecting the pitch.
Meanwhile, winning captain Bavuma, who showed both technique and temperament, had no doubt about the difficulty of the pitch.
"Some balls were bouncing nicely, others were squatting," he said.
"So that was a bit tricky, which made your cross-batted shots a bit harder. But look, I'll always back my defence."
South Africa resumed on 91-7 after a frenetic last session on day two and Bavuma found a staunch ally in Corbin Bosch, who made 25.
Together they added 44 for the eighth wicket to frustrate India on Sunday.
Bavuma, a middle-order batter with tremendous patience, showed terrific application in his 136-ball marathon until he ran out of partners but his knock proved key in his team's narrow win.
"From an experience point of view, if you can be successful in some way as a batter, I think it does a lot of good for your confidence," said Bavuma.
The Proteas will take that confidence into the second and final Test which starts in Guwahati on Saturday as they look to win their first Test series in India since 2000.
