Chasing a tricky 107 on a rain-delayed day, New Zealand lost captain Tom Latham without scoring and Devon Conway for 17.
But Young and Ravindra guided the visitors to 110-2 in the first session and a 1-0 lead in the three-match series.
First-innings centurion Ravindra, a wristy left-hander, looked comfortable as he hit six fours in his 46-ball stay and was named player of the match after the eight-wicket victory.
Young struck the winning boundary, one of his seven fours, as his New Zealand teammates exchanged hugs in the pavilion. He also hit one six, while facing 76 balls.
It was New Zealand's third Test victory in India after their wins in 1969 at Nagpur and at Mumbai in 1988.
"It was extremely difficult. There have been a lot of teams that have come here over that long period of time, so obviously it's a special feeling," Latham told reporters.
"To be in this position, I think the work we did in that first and second innings with the ball and with the bat set up the game for us. A proud moment for this group."
Latham, who took over from Tim Southee after New Zealand were swept 2-0 in Sri Lanka, marked his first Test as full-time captain with a resounding victory.
Fast bowlers Matt Henry and Will O'Rourke set up the win with 15 wickets in the match as India, having chosen to bat after the first day was lost to the weather, were skittled for 46 in their first innings, their lowest home Test score.
"We didn't think we'll be 46 all out but due credit to New Zealand. It set us back and games like this will happen," said India captain Rohit Sharma.
"Got to take the good things forward. We've been here before, conceding home losses, these things happen. We know what it takes and will give it all in the next two Tests."
'Nice having dad here'
On Sunday, India fast bowler Jasprit Bumrah gave the home side a glimmer of hope when he removed Latham lbw with the second ball of the day and New Zealand yet to score.
Bumrah and new ball partner Mohammed Siraj then subjected the New Zealand batsmen to a testing start, with deliveries consistently going past the bat on a lively wicket that had been under covers earlier because of rain.
Bumrah struck a second time when he trapped the left-handed Conway lbw after a successful review to leave New Zealand 35-2.
But that was India's last breakthrough as Young and Ravindra took their team to a famous win.
New Zealand had replied to India's first innings 46 with 402 then dismissed India for 462 in their second innings.
Sarfaraz Khan scored 150 and Rishabh Pant 99 as their 177 stand led India's second-innings fightback, but from 408-3 Henry and O'Rourke sparked a collapse as the hosts lost their last seven wickets for 54.
Ravindra, a Wellington-born batting all-rounder with roots in Bengaluru and his father watching from the stands, hammered 134 in New Zealand's first innings.
"Whenever I play in Bangalore it's quite nice to have the support," said Ravindra.
"My phone starts blowing up. The fans are great too. I love the support. It's nice having dad and close family here, it always makes it special."
New Zealand came into the match without leading batsman Kane Williamson, who is recovering from a groin injury.
The second Test begins on Thursday in Pune. The third is on November 1 in Mumbai.
