Williamson along with the stodgy Tom Blundell (30, 80 balls) added 47 runs before the lanky speedster, playing his 97th Test, bowled one on length, which breached the opener's defence.
New Zealand went into the final session with only 50 runs required to surpass India's modest first innings total.
Williamson looked good as he hit some delightful strokes square off the wicket. The square drive on the rise off Jasprit Bumrah (0/37 in 11 overs) followed by a cover drive showed his class.
Bumrah, in particular, was punished by Williamson, who also back-cut him for a boundary and Taylor then punished another half volley through the covers.
There were quite a few loose deliveries on offer from the Indian pacers and in between a few did beat the bat. With the 'Basin' baked in sunshine, batting became lot more easier and the Black Caps seized the initiative.
Shami (0/34 in 9 overs) and Bumrah didn't exactly look menacing during that second session, while Ravichandran Ashwin (0/26 in 9 overs), despite being hit for a six, showed a big heart in bowling attacking lines as he got the deliveries to drift.
Earlier, New Zealand's debutant Kyle Jamieson and veteran Tim Southee took four wickets apiece as Indian innings folded in 68.1 overs.
Jamieson (4/49 in 16 overs) and Southee (4/49 in 20.1 overs) took four of the five wickets that fell on the second morning with India adding only 43 runs to their overnight score of 122 for 5.
Rishabh Pant (19) started with a six but then a horrible mix-up with senior partner Ajinkya Rahane (46) resulted in a run-out and the little chance of recovery was gone for good.
Ravichandran Ashwin got a beauty from Southee, pretty similar to what Prithvi Shaw got, while Rahane inside edged one while trying to leave it alone.
With India reduced to 132 for 7, Rahane knew that time was running out as he played a square drive off Trent Boult to get a boundary.
Southee got rid of Rahane soon when he tried to shoulder arm a delivery that made a late inward movement.
Mohammed Shami's entertaining 21 then enabled the visitors to cross the 150-run mark.
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