Set a target of 339 runs, a collapse-prone Bangladesh were all out for 123 in 29.3 overs in their second innings in the second session of the third day, without offering much of a challenge.
Dananjaya finished with 5-24 to take his match haul to eight wickets while experienced Rangana Herath claimed 4-49.
Sri Lanka bowled out Bangladesh for 110 runs in the first innings to take a 112-run lead after they made 222 runs in their opening innings.
The visitors’ victory was never in doubt after they posted 226 runs in their second innings, resuming the third day’s play on 200-8.
Roshen Silva remained unbeaten on 70, adding 48 runs with fellow overnight batsman Suranga Lakmal for the ninth wicket to extend their lead on a difficult pitch at the Sher-e-Bangla National Stadium.
Left-arm spinner Taijul Islam bowled Lakmal for 21 to end the partnership before he trapped Herath leg-before off the next ball to wrap up the Sri Lanka innings.
Taijul finished with 4-76 taking his tally to eight wickets in the match but Bangladesh were always in the back foot once they started their second innings on a massively turning wicket.
Sri Lanka began their victory push as Dilruwan Perera removed opener Tamim Iqbal for two in just the second over of the innings.
Herath got rid of Imrul Kayes before the lunch break as wicketkeeper Niroshan Dickwella took a catch, just a ball after he hit the left-arm spinner for a six.
Herath added Mominul Haque (33) and Mushfiqur Rahim (25) to after the lunch break before Dananjaya stepped on to the scene and did the rest.
Silva, who also scored 56 runs in the first innings, was man of the match.
The first Test ended in a draw in Chittagong.
The two teams will now play two Twenty20 internationals with the first match in Dhaka on February 15.
The West Indies had originally been scheduled to tour Pakistan at the end of 2017 but thick smog cover and a reluctance on the part of several senior players had seen the series being postponed indefinitely.
Sethi has said that the three matches will be held at the end of March at the Gaddafi stadium in Lahore which was also the venue for the historic T20 series with an ICC World XI in September last year.
Along with the confirmation of the Caribbean side’s visit, Sethi also said that he has signed a trilateral series involving Pakistan, West Indies and Bangladesh to be held in August 2016.
The series has been proposed to take place in the United States of America with Florida and Houston being touted as some of the likely venues.
This will be the first time that the Pakistan cricket team will play on American soil. West Indies have already had a taste of playing in the global superpower when the T20 matches against India last year were held in the States.
The hosts ran out comfortable victors by five wickets in the clash at Hobart on Wednesday after having thrashed New Zealand in the opening match of the series.
We look at the three key players to watch out for as the old foes meet once again.
After being called out by Steve Smith publicly for his training intensity, it seemed that the doors to the national team had all but shut on the floundering maverick batsman.
Having found a spot in the recently concluded ODI series due to injuries, the often derided Glenn Maxwell seems to have found a new lease of life in the T20Is. He answered all his critics with a man-of-the-match performance in the previous match where he registered his second T20I ton while also chipping in with three crucial English wickets with his part-time off spin.
It will be interesting to note if Maxwell can kick on from the superb ton or whether it ends up as one of the many false dawns in his career.
The Hobart Hurricanes opening batsman smashed the Big Bash League (BBL) record for the most runs scored in a single edition just recently. His terrific displays have seen him earn a mouthwatering IPL contract with Rajasthan Royals along with a maiden Australia cap in the T20I series.
He looked extremely good on his debut on Wednesday where he launched two massive sixes and as many boundaries in a 20-ball stay at the crease which yielded 30 runs. Short will want to build on that promising start as he looks to tie down a spot in the national team for the foreseeable future.
The English all-rounder was brilliant with the ball against the Aussies in Hobart as he picked up three wickets. He failed to do much with the bat though and will be looking to make amends in Melbourne.
He was in sparkling form himself in the recently concluded BBL and made head turns when he tore apart Nathan Lyon in England’s warm-up T20 clash against a Cricket Australia XI. The southpaw smashed Lyon’s first five deliveries for sixes before finding a boundary off the last ball as 34 runs came in a single over.