Santner, who replaced legspinner Ish Sodhi, returned figures of 2-24 and Patel claimed 2-32 to rip through the top order after the hosts won the toss and elected to bat in pursuit of a first ever Test series win over the Black Caps.
New Zealand's bowling attack maintained a tight line and length to create pressure by conceding just eight runs in the first seven overs on Wednesday.
Bangladesh's first boundary came in the eighth over when Zakir Hasan drove Patel through cover for four.
The pressure eventually got the better of Zakir (eight) when he played a rash shot against Santner and chipped it straight to mid-on.
In the next over, Patel dismissed Mahmudul Hasan (14) after he was beaten by the drift to give a catch at short-leg.
Mominul Haque (five) was the next to go when he was caught behind off Patel's bowling.
But the biggest blow for Bangladesh followed in the next over when captain Najmul Hossain Shanto (nine), who hit a century in the first Test, attempted a reverse sweep against Santner.
The on-field umpires turned down the appeal but the tourists reviewed it successfully to leave Bangladesh 4-47.
Mushfiqur Rahim (18) and Shahadat Hossain (14) combined to see off the remainder of the session without further trouble.
Bangladesh lead the two-match series after winning the opener by 150 runs and retained the same starting line-up for the second Test.
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