New Zealand openers Tom Latham (2) and Devon Conway (1) survived a tricky three overs before lunch on day four to reach 0-4.
Patel's decisive bowling was complemented by left-arm spinner Mitchell Santner's 3-51.
Zakir Hasan battled alone for Bangladesh in the second innings with 59 off 86, hitting six fours and one six.
Resuming at 2-38, Bangladesh started strongly with Zakir Hasan hitting skipper Tim Southee for a pair of boundaries in the second over of the day.
But Patel got into the act, trapping Mominul Haque lbw for 15, ending a 33-run partnership between him and Zakir, which was Bangladesh's best partnership in the second innings.
Bangladesh then collapsed and lost the rest of the seven wickets for 73 runs, bolstering New Zealand's chance to level the series.
Santer dealt a double blow, dismissing Mushfiqur Rahim (9) and Shahadat Hossain (4) in his consecutive overs, leaving Bangladesh at 8-55.
Mushfiqur's hesitation to play or leave the quicker delivery of Santer caused his dismissal.
Then Mehidy Hasan played a rash shot against Patel to throw his wicket away to further the home side's slide.
Bangladesh, dismissed for 172 in their first innings, won the first Test by 150 runs.
