Taylor and Henry Nicholls kept the runs flowing with a 216-run stand, during which both batsmen surpassed their hundreds, before Nicholls was clean bowled by Taijul for 107, which came off 129 deliveries and included nine boundaries.
The final test starts on Saturday in Christchurch, although New Zealand coach Gary Stead said they would be cautious with Williamson, who is likely to be replaced by the uncapped Will Young.
Bangladesh have just begun their chase and are now four for one.
This win by the Kiwis can be considered more impressive considering the fact that rain washed out the first two days of play.
"Will has been in our Test squad as batting cover for the past two series and he should be confident following some strong performances domestically and for New Zealand A".
The New Zealand bowlers were relentless but Bangladesh batsmen also didn't cause their help by playing some wretched shots that led to their crushing innings and 12 runs defeat to New Zealand at Basin Reserve in Wellington on Tuesday.
Three of the dismissals, including the key scalp of Tamim Iqbal for 74, came in a sharp 13-ball spell from Wagner either side of lunch.
Ross Taylor moved past Martin Crowe for the second most Test centuries for New Zealand as the Black Caps dominated Bangladesh in Wellington.
Taylor and Williamson carried New Zealand to 38-2 at stumps on day three and gradually played New Zealand into a position on day four from which they have some hope of winning the match and clinching the three-test series.
In the face of Wagner's relentless bouncer barrage, Bangladesh were all out for 209 to give New Zealand victory by an innings and 12 runs to go with their innings and 52-run win in the first Test. He helped New Zealand negotiate a hard first half-hour Monday when the ball still dominated the bat and went on to reach his half century from 67 balls. They then received a lesson in seam bowling technique for Bangladesh's Abu Jayed who hit a ideal length, made the ball duck about and dismissed New Zealand openers Tom Latham for 4 and Jeet Raval for 3.