New Zealand, who collapsed for 153 in their first innings, lost opener Tom Latham for a duck when they returned to bat in the final session but recovered to reached 56-1 at stumps.
Resuming at 59 for two, Pakistan's overnight batting pair Azhar Ali and Haris Sohail made steady progress in the morning session to add 64 runs before falling in consecutive overs.
Sohail (38) chipped a full-toss off spinner Ish Sodhi to mid-wicket before Boult forced an edge off Ali (22), which was matched with a stunning one-handed catch by wicketkeeper BJ Watling.
Azam and Asad Shafiq batted with intent as they improved their scoring rate to reach 144-4 at lunch.
Pakistan were threatening to run away with the game, having moved into the lead, but the run flow was choked by fast bowler Neil Wagner, who at one point racked up four consecutive maidens.
It was Boult who provided the breakthrough when Shafiq (43) could only chop a searing inswinger back on to his stumps.
Patel, at the other end, was rewarded for his extended spell as he removed Pakistan captain Sarfraz Ahmed through a mistimed sweep to square leg for his first test wicket.
The 30-year-old Indian-born Kiwi did not have to wait long for his second wicket, with Bilal Asif getting himself stumped off a flighted delivery.
Yasir Shah showed signs of resistance but, after playing out 38 balls for nine runs, became the 150th test scalp for Wagner.
The Kiwis wrapped up the innings when Azam feathered a Boult delivery down the leg side to Watling.
Azam finished with a gritty 62 off 109 balls to give the hosts a precious lead on a deteriorating surface, while Boult finished with an impressive 4-54.
The visitors exuded calm in the final hour after losing Latham in the second over as skipper Kane Williamson (27) and Jeet Raval (26) trimmed Pakistan's lead to 18 runs, negotiating a disciplined opening spell from in-form quick Mohammad Abbas.
(Reporting by Hardik Vyas in Bengaluru)
