With the win, Bangladesh sealed the three-match series 2-0 on Wednesday.
Litton hit a career-best 83 off 41 balls and Rony Talukdar made 44 off 23 as the two openers guided Bangladesh to 202-3 in a rain-reduced 17 overs.
Shakib then claimed 5-22 to restrict Ireland to 125-9, overtaking New Zealander Tim Southee‘s 134 wickets in T20Is.
Litton, who hit 10 fours and three sixes, reached his fifty off 18 balls, the quickest T20 fifty for Bangladesh. He surpassed Mohammad Ashraful’s 20-ball half-century against the West Indies in 2007.
Curtis Campher top-scored for Ireland with 50 but it came much too late.
The visitors were never in the contest after Taskin Ahmed, who finished with 3-27, claimed a wicket off the first ball of the innings. Shakib then reduced Ireland to 43 for 6.
Litton and Rony shared 124 runs in their opening stand, the best start by Bangladesh in a T20, after Ireland captain Paul Stirling won the toss and elected to bowl.
Leg-spinner Ben White earned Ireland their first breakthrough when he forced Rony to give a catch to Mark Adair at long on.
White, the only Ireland bowler to give Bangladesh any trouble, also dismissed Litton in his next over before finishing with 2-28.
However, Shakib (38 not out) and Towhid Hridoy (24) put on 61 runs for the fourth wicket to help Bangladesh put on a big total.
“That was a tricky day, no doubt about that… There were probably a few of the same errors from the previous couple of weeks. So a hell of a lot to work on,” Stirling said.
Bangladesh won Monday’s opening game by 22 runs. The third match is scheduled at the same venue on Friday.
The home side also won the preceding three-match one-day international series 2-0.
Bangladesh “are a strong team wherever they play in the world”, Stirling said.
“We knew (the tour) was going to be tough. We probably didn’t think it was going to be as one-sided as it has been,” he said.
(With AFP inputs)