At tea, the hosts reached 3-111 with opening batter Mayank Agarwal unbeaten on 52 and Shreyas Iyer on seven not out.
Agarwal and Shubman Gill gave their side a solid start after captain Virat Kohli had won the toss and opted to bat.Â
The first session was wiped out due to a wet outfield after unseasonal rain in Mumbai over the past two days.
Mumbai-born Patel, whose parents immigrated to New Zealand in 1996, dismissed Gill for 44 when he was caught in the slips after the batter had added 80 for the opening stand with Agarwal.
Patel returned in his next over to dismiss Cheteshwar Pujara and Kohli before they had opened their accounts, much to the disappointment of the few hundreds present at the ground which overlooks the Arabian Sea.
Kohli, who returned to the side to lead India after being rested for the three-match Twenty20 series and the first Test in Kanpur, reviewed the lbw decision unsuccessfully and trudged off after a chat with the on-field umpires.
Television replays were inconclusive in showing if he had hit the ball before Patel's delivery struck his pad.
From 80 for no loss, India were reduced to 3-80 and it was left to Agarwal and Iyer, who made a dream Test debut in Kanpur with a first-innings 105 followed by a crucial 65 in the second, to ensure they had no further setbacks.
Both sides were dealt injury blows before the Test with the tourists losing captain Kane Williamson to an elbow problem while the hosts lost Ajinkya Rahane, who led the side in Kanpur in Kohli's absence, Ravindra Jadeja and Ishant Sharma.