Nadal needed all his experience to keep Pouille at bay in the decider and the match looked to be heading for another tiebreak before he pounced to break Pouille's serve for the first time to go 6-5 up.
The Spaniard held his nerve to close out the match in clinical fashion after two hours and 31 minutes and will next face Russian Karen Khachanov, who beat Chinese wild card Wu Di 6-4 6-2.
"(Pouille) played well. Very aggressive, He's serving well," Nadal said. "For me it was little bit difficult at the beginning. Then I started to play better. But still I didn't have the control of the match for almost all the time.
"I remember the match against him in the U.S. Open... That time was for him, today was for me."
Second seed Alexander Zverev served seven aces as he beat Britain's Kyle Edmund 6-3 7-6(3) in the day's last match and will meet Italy's Fabio Fognini in the next round.
Juan Martin del Potro stormed into round two with a 7-6(4) 6-4 win over Uruguay's Pablo Cuevas and will face third seed Grigor Dimitrov, who battled past Bosnian Damir Dzumhur 6-1 3-6 6-3.
Eighth seed Nick Kyrgios breezed past Georgia's Nikoloz Basilashvili 6-1 6-2, the Australian serving 12 aces as he closed out the contest in under an hour.
John Isner, the number six seed, also pulled off a routine 6-2 6-3 victory over Tunisian qualifier Malek Jaziri, although two of the American's unseeded compatriots bowed out.
Jared Donaldson fought hard against Czech seventh seed Tomas Berdych, but ran out of steam as he went down 6-3 0-6 6-2.
Jack Sock was also eliminated and the 25-year-old will be disappointed he could not hold off Andrey Rublev despite comfortably winning the first set against the Russian, who bounced back to win 3-6 6-1 6-2 and will face Berdych next.
(Writing by Simon Jennings in Bengaluru, editing by Ed Osmond and Ken Ferris)
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