Ireland survived an inspired spell of swing bowling from Zimbabwe's Tinashe Panyangara to scrape through for a three-wicket win on the last ball of their group B encounter in Sylhet.
Zimbabwe captain Taylor (46-ball 59) hit six fours and two sixes and anchored Zimbabwe's batting as his team posted 163 for five in their twenty overs after being put in to bat.
Elton Chigumbura hit 22 off 13 balls with two sixes in the last over to take Zimbabwe to a competitive total after Irish spinners George Dockrell (2-18) and Andy McBrine (2-26) restricted them in the middle overs.
Chasing 164 for a St Patrick's Day victory, the Irish nation seemed to be cruising against their Test-playing opponents after opener Paul Stirling's 34-ball 60.
Stirling hit nine boundaries and a six and with captain William Porterfield (31) adding 80 runs for the opening wicket, but Panyangara's double strike in the 15th over brought Zimbabwe back in the game.
Panyangara (4-37), who was taken for 18 in his first over by Stirling, also dismissed the dangerous Kevin O'Brien (17) in the 18th over and returned to bowl the final over of the innings with Ireland needing just four runs for victory.
In the final over, Ireland lost Ed Joyce (20) to Panyangara and Max Sorensen to a run out off successive balls but managed to scamper for a bye off the last delivery for the winning run with wicketkeeper Brendan Taylor missing a run out.
Nepal's instant impact
Earlier, Nepal claimed an impressive 80-run victory against Hong Kong in the battle of the tournament debutants on the opening day of the World Twenty20.
Gyanendra Malla (48) and Paras Khadka (41) shared an 80-run stand for the third wicket to underpin a total of 149-8 after Nepal had been asked to bat in the Group A fixture in Chittagong.
Hong Kong were bowled out for 69 in reply with left-arm spinners Basant Regmi (3-14) and Shakti Gauchan (3-9) doing most of the damage.
The first six days of the three-week tournament are devoted to reducing eight teams to two, with the respective winners of Group A and B advancing to the second phase.
Hosts Bangladesh beat Afghanistan in the tournament opener in Group A, while Group B gets under way on Monday in Sylhet with Ireland v Zimbabwe and Netherlands v United Arab Emirates.
The second group stage starts on Friday with India v Pakistan in Dhaka.
Meanwhile, Avinash Karn has been ruled out of the World Twenty20 with an injured right knee and has been replaced by Anil Kumar Mandal in Nepal's 15-man squad.