"This was my first match in quite a while, so I'm really happy to have had such a competitive match against such a great player," Konta said.
"It was a really tough first round for both of us. Madison is an incredibly good player, so I feel very lucky that I got to play her so early on and really have a good match."
World number 19 Keys missed eight opportunities to break Konta's serve before she finally succeeded while up 5-4 to take the first set with a powerful forehand drive.
Konta appeared to calm her serve and nerve in the second set, fighting back with an early break to go up 3-2. The world number nine then held on to level the match at one set apiece.
The Briton battled her way to a 2-0 lead in the decider as Keys began to tire and finished the match with another break to set up a clash with Ajla Tomljanovic in the next round.
Belarusian qualifier Aliaksandra Sasnovich came from behind to upset sixth seed Kristina Mladenovic 1-6 6-3 7-5 while Elina Svitolina beat Spain's Carla Suarez Navarro 6-2 6-4. Ukraine's Lesia Tsurenko then stunned eighth seed Ashleigh Barty 6-3 6-2.
Double Wimbledon champion Petra Kvitova withdrew from the event with a viral illness. The 27-year-old Czech was unable to take the court against Estonian Anett Kontaveit and was replaced in the draw by lucky loser Heather Watson.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Christian Radnedge)
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