In reply to Afghanistan's 153-7, Dilshan smashed eight boundaries and three sixes in his unbeaten 83-run knock to help Sri Lanka chase down the target with seven balls to spare.
Meanwhile, skipper Angelo Mathews also chipped in with significant 21 in his side's innings.
Earlier, Afghanistan made a slow start as openers Mohammad Shahzad (eight) and Noor Ali Zadran (20) got dismissed without making much of an impact on the innings.
It was skipper Asghar Stanikzai who produced a brilliant batting display, smashing 62 off just 47 balls to strengthen his side and post a modest target of 154 runs against the opponents in their stipulated 20 overs.
Samiullan Shenwari (31) was the other notable figure for Afghanistan.
In contrast, Thisara Perera was the pick of the bowlers for Lanka as he scalped three wickets while Rangana Herath added two wickets in his account.
Angelo Mathews and Nuwan Kulasekara also gobbled up two wickets.
Dilshan was adjudged `Man of the Match` for his dazzling contribution with the bat.
Lanka will next face West Indies in their group match at M.Chinnaswamy Stadium in Bengaluru on Sunday.