With Shanto unbeaten on 104, the home side finished day three on 3-212 for a lead of 205.
With seven wickets in hand they look well placed to go 1-0 up in the two-Test series.
New Zealand, who made 317 in their first innings, face a daunting task batting last on a deteriorating pitch in the spin-dominated contest.
Shanto, who recorded his third hundred in four Test innings, was still going strong when bad light stopped play with Mushfiqur Rahim on 43 at the other end.
New Zealand had resumed on Thursday on 8-266 and went on to eclipse Bangladesh's first-innings total of 310 after Tim Southee (35) and Kyle Jamieson (23) put together a 52-run stand for the ninth wicket.
Bangladesh looked in trouble shortly after lunch when they lost both their openers early in their second innings.
Zakir Hasan (17) fell to Ajaz Patel for the second time in the match while Mahmudul Hasan Joy's stay was cut short in bizarre fashion.
Shanto's straight drive brushed bowler Southee's fingers before hitting the stumps at the non-striker's end with Joy (eight) short of the crease.
But there was to be no collapse, as Shanto combined with Mominul Haque (40) in a fluent 90-run partnership to smooth any fears of a crisis.
Mominul was run out after an almighty mix-up with Shanto, who went on to consolidate Bangladesh's position with a 96-run partnership with Mushfiqur for the unbroken fourth wicket.
