Munro, in the side as one of four Black Caps changes, smashed 85 and put on 130 with Taylor, who accelerated dramatically thereafter to make 107 not out.
Bangladesh arrived in Fatullah with a 2-0 lead in the three-match series but face a tough task to make it a clean sweep, especially with opener Tamim Iqbal sidelined by abdominal pain.
Both the preceding Tests were drawn, while a one-off Twenty20 international follows on Wednesday.
Anton Devcich hit five fours in six balls faced in the second and third overs as he and fellow opener Tom Latham brought up a 50 partnership in less than eight overs.
He added three more before falling four short of a half-century when he top-edged a sweep at Mahmudullah and was caught by Abdur Razzak to end a stand of 66.
Latham punished Mushfiqur Rahim for a missed stumping off Razzak by hitting the spinner for six, although Grant Elliott holed out two balls later for three.
Latham followed for 43, made from 73 balls with two sixes but no fours, when he edged Rubel Hossain to Mushfiqur.
Taylor had made just 11 when Rubel dropped him off his own bowling and he and Munro brought up their 50 stand in 10.3 overs, the ball after Munro reverse-pulled Razzak for six.
He launched another maximum off Sohag Gazi and reached a 53-ball half-century with a four off Mashrafe Mortaza.
Back-to-back boundaries took Taylor into the 40s and he followed his partner to 50 moments before Munro gloved Mahmudullah to Mushfiqur, again attempting a reverse hit.
Corey Anderson scored just one but Taylor hit two fours off Razzak, a mere prelude to three consecutive sixes off Gazi as he powered his side towards a big total.
He added three fours in a Mortaza over and reached his hundred with two balls remaining as the Kiwis hit 98 off the last 10 overs.
Taylor's first 50 runs came off 69 balls with two boundaries but after Munro's departure he added 57 from a further 24, with seven more fours and those three maximums.