Nair announced his arrival at the world stage with an unforgettable triple ton (303 not out) as India broke a plethora of records en route their highest ever Test total of 759 for seven against England on the penultimate day of the fifth and final match.
"Congratulations on the historic triple century @karun126! We all are delighted & proud of your remarkable feat," Modi wrote on his twitter handle.
"what a player! What a knock! Take a bow, @karun126 for being 3rd cricketer & 1st Indian to convert maiden century into a triple hundred!" tweeted Rathore.
Nair notched up the milestone in only his third Test and first series.
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"Congratulation Karun for this amazing feat. Your batting was like poetry. I hope you get another one very soon," Sehwag, who is the only other Indian to have scored a triple ton, said.
He also wrote on his twitter handle: "Yay ! Welcome to the 300 club @karun126 . It was very lonely here for the last 12 years 8 months. Wish you the very best Karun.Maza aa gaya!"
VVS Laxman said it was great to see that Nair was still hungry for runs even after completing his century.
"Rahul Dravid would be proud of him. He has taught players to work hard and stay committed to their game. Karun spent two months with Dravid during India A tour of Australia.
"He probably went through mental conditioning during that period. He showed the hunger to keep going even after completing his hundred," said Laxman.
Board CEO Rahul Johri said, "Many congratulations @karun126, what a magnificent knock @bcci #IndvsEng."
Mohammad Kaif said, "Karun Nair reaches 250, a landmark only 3 other Indians @virendersehwag (4times), Dravid and @VVSLaxman281 (1each) have reached."
Tennis star Mahesh Bhupathi retweeted Sehwag's tweet, calling it a "Classic".
India spinner Pragyan Ojha said, "Staying at the wicket & scoring big is all about temperament. Full credit to @karun126 for showcasing that."
Ajinkya Rahane said, "Well done karun @karun126 very well deserved. Keep it going."
West Indies' big-hitting opener Chris Gayle said, "Welcome to the 300 club youngster @karun126 ?????????? #Super."
Legendary Australian spinner Shane Warne wrote, "In bed watching India V England. Great batting from Karun. But India should've declared 40 mins ago & had an hour at England.