The meeting came as divisions grow in Europe over the proposed tariffs
Junaid Siddique hit a solid 106 for his maiden Test century and Mushfiqur Rahim an impressive 95 to frustrate the tourists before Bangladesh were bowled out for 331 on the fifth and final day, chasing an improbable 513-run target.
Off-spinner Swann, who took five wickets in the first innings, finished with 5-127 off 49 overs in the second.
England, denied success for more than two sessions by left-handed Siddique and wicket-keeper Rahim, heaved a sigh of relief when Swann removed both batsmen in his opening four overs after lunch.
Swann struck in his first over after the break when he had Siddique caught by Paul Collingwood in the slips before accounting for Rahim, who was bowled after he stepped out to drive the spinner and missed the line.
Rahim, who came in to bat after five wickets fell for 110 before the tea break on Monday, was involved in a 167-run stand for the sixth wicket with Siddique before missing out on his second Test hundred.
Siddique hit 17 fours in his 292-ball knock and Rahim 12 in a 232-ball innings.
Bangladesh’s innings lasted only 10 more overs after the dismissals of Siddique and Rahim, with paceman Stuart Broad and Tim Bresnan, and Swann sharing the last three wickets.
Siddique and Rahim kept the England attack at bay in the morning session with their determined knocks, helping their team add 86 to their overnight total of 191-5.
Skipper Alastair Cook took the new ball after 80 overs, but his pacemen Broad, Bresnan and debutant Steven Finn failed the unsettle the Bangladeshi pair.
Siddique, 68 not out on Monday, completed his hundred in style as he drove Finn through the covers for a four to reach 96 and then smashed the same bowler through mid-off for another boundary.
Rahim, who made 79 in the first innings, completed his second successive half-century early in the morning when he drove Finn through the covers for a four.
Siddique offered a difficult chance on 106 when he attempted to cut Swann, but wicketkeeper Matt Prior failed to hold on to it.
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