Bangladesh's batting never recovered after losing openers Tamim Iqbal and Shamsur Rahman in the first two overs.
Senanayake broke Bangladesh's back with the wickets of Shakib (12) and Anamul in successive overs to reduce the hosts to 41-4 in the sixth.
It soon became 59-6 as Ajantha Mendis removed Nasir Hossain and Mahmudullah was run out.
Sabbir and Mortaza, who hit 17 off 10 balls, took Bangladesh past the 100-run mark.
The two teams will begin a three-match one-day series in Dhaka on Monday.
Having bowled the hosts out for a paltry 120, Sri Lanka stuttered in their reply to lose six wickets for 50 runs by the 12th over at the Zahur Ahmed Chowdhury stadium.
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Veteran Kumar Sangakkara and Thisara Perera rebuilt for the world's top-ranked Twenty20 side with a 46-run partnership for the seventh wicket.
But when Sangakkara was snapped up at point off Rubel Hossain for 37, Sri Lanka still needed 25 to win off the last 20 balls.
Senanayake helped Perera (35 not out) i