India went into the matches as the world's leading Test cricket side. However they carried the burden of never having won in Australia.
"A historic cricketing accomplishment in Australia!" said Modi on his official Twitter handle. "Congratulations to the Indian Cricket Team for the hard-fought and richly deserved series victory. "The series witnessed some memorable performances and solid teamwork. Best wishes for the various games ahead."
Former cricketer Sachin Tendulkar called it an amazing day for Indian cricket.
"#TeamIndia has made this a series to be cherished. Proud of the boys," he wrote on his social media account.
The legendary batsman singled out Cheteshwar Pujara for his 521 runs in four matches - including 193 in the first innings in Sydney. They were performances that earned him a man-of-the-series accolade.
"Test match batting at its best by Pujara," added the 45-year-old who retired in 2013.
"@cheteshwar1 has been the difference between the 2 teams for me. To be able to bat for such long periods of time is a testament to his concentration and understanding of the game." Tendulkar wrote on Twitter.
Kohli highlighted the success of a dedicated squad. "So proud to be a part of this team, great feeling! This is not just a team this is a family. Onwards and upwards from here on," he said on Twitter.
"The boys make the captain look good. By far, this is my biggest achievement. It's at the top of the pile."
His Australia counterpart, Tim Paine, was munificent in defeat. "We have to tip our hat to India," he said. "We know how tough it is to win in Australia so congrats because it's a huge achievement."
Paine added: "I am disappointed, definitely over the last two Tests. We had our chances in Adelaide, in Perth I thought we played some good cricket, but in Melbourne and Sydney, we've been outplayed."
India launched their tour of Australia by winning the opening Test in Adelaide by 31 runs on the fifth day. It was their first Test win in Australia for a decade.
The victory was built on Pujara's first innings 123 and his second innings 71, while wicketkeeper Rishabh Pant took a world record-equalling 11 catches. Australia bounced back in Perth to level the series, winning by 146 runs.
It was their first Test victory since the ball-tampering scandal in March that led to the suspensions of three of their star batsmen including skipper Steve Smith and vice-captain David Warner.