Rehman ended the resistance of Mushfiqur Rahim (49) an hour into the second session on the fourth day, finishing with figures of four for 88 after he had earlier dismissed debutant Nazimuddin and former captain Shakib Al Hasan.
Nazimuddin and Shakib both completed half-centuries before throwing away their wickets, allowing Pakistan to quicken their charge toward victory.
Saeed Ajmal bowled Mahmudullah for a duck and Aizaz Cheema took the final wicket when he had paceman Shahadat Hossain caught by Umar Gul at mid-off for 21.
Bangladesh finished the second innings on 275-9 as last man Rubel Hossain was unable to bat after injuring his shoulder while trying to save a boundary the previous day.
"Credit goes to everyone, they are really putting in the effort," Pakistan captain Misbah-ul Haq said in a news conference.
"They are really up to the mark, they are really fulfilling their responsibility, that's why the team is winning," he added.
The home side began the day with little hope of saving the match and suffered their first blow in the 10th over when Rehman had Shakib for 51 soon after the all-rounder reached his ninth Test 50 with a single off the same bowler.
Nazimuddin, on 41 overnight, also survived some anxious moments before he completed his maiden Test half-century with three runs off a Rehman delivery.
Wicketkeeper Adnan Akmal dropped him twice on 51 and 57 but the right-hander was unable to take full benefit of his reprieves.
Two sixes in consecutive overs, off Rehman and Ajmal, seemed to have given him his confidence back but the local boy finally paid the price for his aggression when Ajmal took a catch off Rehman at long off, ending Nazimuddin's innings on 78.
Elias Sunny was unbeaten on 20 when the match came to an end with more than a day and a session to play.
Bangladesh captain Mushfiqur Rahim was left ruing their performance of opening day.
"I think the first two sessions on the opening day cost us the game," he said.
"We did not do that bad after that period. If we could survive those two sessions, may be the situation could have been different," he added.
Bangladesh were bowled out for 135 runs in their first innings on the first two sessions of the opening day before Pakistan batted more than five sessions to declare their innings on 594-5 on the third day.
