Zimbabwe were trailing by 259 at lunch as they seek revenge for an innings and 106-run loss in Bangladesh early last year.
Taylor is deputising for regular skipper Sean Williams, who is in self-isolation after coming into contact with family members who had tested positive for coronavirus.
The captain averaged close to a run a ball as he faced 92 balls and struck 12 fours and one six before being caught by substitute Yasir Ali off the bowling of Mehidy Hasan Miraz.
Taylor shared a 115-run second-wicket partnership with opener Takudzwanashe Kaitano, who remained unbeaten on 63 with his 200-ball stand including eight fours.
Mehidy and fellow bowlers Taskin Ahmed, Ebadot Hossain, Shakib al Hasan and captain Mominul Haque toiled in warm mid-winter conditions on the third day, conceding 95 runs and capturing just one wicket.