Kohli had returned home to attend the birth of his daughter after the opening test loss, leaving behind a deflated side who were skittled out for 36 in Adelaide - their lowest innings score in 88 years of test cricket.
Under stand-in skipper Ajinkya Rahane, India levelled the series in Melbourne and drew in Sydney before triumphing in Brisbane to retain the Border-Gavaskar trophy.
"WHAT A WIN!!! Yessssss. To everyone who doubted us after Adelaide, stand up and take notice," Kohli tweeted after India's three-wicket victory at the Gabba.
"Exemplary performance but the grit and determination was the standout for us the whole way. Well done to all the boys and the management. Enjoy this historic feat lads."
Modi also took to Twitter to congratulate the team, who showed great resilience despite losing their entire frontline attack to injuries heading into the Brisbane decider.
"We are all overjoyed at the success of the Indian Cricket Team in Australia. Their remarkable energy and passion was visible throughout. So was their stellar intent, remarkable grit and determination..." he said.
The Indian cricket board announced a $960,000 cash award for the team.
Former England captain Michael Vaughan, who was among those who had predicted a 4-0 humiliation for India, hailed what he called "one of the greatest if not the greatest Test victory of all time.
"...Egg on my face over here in the UK .. but I love to see character & skill .. India have it in abundance," the 46-year-old tweeted, identifying India opener Shubman Gill and stumper-batsman Rishabh Pant as "future superstars".
Ex-Australia captain Ricky Ponting, who had made a similar prediction, was disappointed with the performance of the hosts.
"The cold hard facts of it are pretty much that was the India A team that played this test match and (India) still won," Ponting told www.cricket.com.au.
"That's a real concern for Australia."