Sindhu outclassed Japan's Nozomi Okuhara 21-19 21-10 in the semi-final to assure herself of at least a silver medal.
Bindra, India's lone Olympic individual gold medal winner, took to twitter to congratulate the Hyderabadi shuttler.
"What a player !! @Pvsindhu1 I'm waiting for you to join me in the club. You have no idea how lonely it's been !!!," Bindra tweeted.
"Way to go Sindhu! Congratulations! You have made India proud!!! All the very best for the finals," the PM wrote on his official handle.
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"Superb performance @PvSindhu1. You make India proud! Best of luck for the finals," read another tweet from Modi.
London Olympics bronze medallist shooter Gagan Narang also joined fellow shooter Bindra in congratulating Sindhu.
"Super #PVSindhu nails #Rio2016 medal. May grey turn to yellow for you super girl! #GoYellow @Pvsindhu1 @pullelagopicha1," Narang wrote.
Bollywood Superstar Amitabh Bachchan was not behind.
"T 2352 - Never ever underestimate power of female gender ! #PVSindhu you have destroyed so many 'naysayers' .. You are the PRIDE of INDIA !," the mega star wrote.
Star boxer Vijender Singh had his own style of eulogising Sindhu.
(REOPENS SPD 16)
Sindhu's father P V Ramana was speechless and credited coach Pullela Gopichand.
"Very, very happy no words to describe. From the age of 10 Gopi has taken care of her. We are very grateful to him and Sindhu has also struggled to win an Olympic medal.
"I pray God to give her good health. I wish she wins so no one can say anything," Ramana said.
Her mother P Vijaya said, "I want to congratulate Sakshi for winning bronze and Sindhu, we are very happy for her victory.