His effort helped the islanders surpass Afghanistan's 153 for seven with seven balls to spare at the Eden Gardens Stadium in Kolkata.
"In my last two games I had two ducks. I needed runs going into the big games. I'm really happy that I finished the game," Dilshan, 39, said after being named man of the match.
"In these kinds of tournaments, the first match is very important. Senior guys need to stand up and do it for the team (so that) the middle order can bat around us."
Dilshan gave Sri Lanka a flying start, smacking fast bowler Dawlat Zadran for two sixes on the trot, including one that came off his famous 'Dilscoop' shot which typically sails over the wicketkeeper's head.
Offspinner Mohammad Nabi briefly put the brakes on Sri Lanka by dismissing opener Dinesh Chandimal for 18, before 17-year-old legspinner Rashid Khan accounted for Lahiru Thirimanne (six).
Wicketkeeper Mohammad Shahzad then sent back Thisara Perera with a direct throw at the stumps as Sri Lanka slipped to 85 for three, but Dilshan kept his cool to ensure a smooth finish for his side in the end.