Haq's 13th half-century, his 11th in 13 Tests as captain, had kept Pakistan in a good position after openers Mohammad Hafeez had made 88 and Taufiq Umar 58 on a slow turning Dubai Stadium pitch.
But Swann trapped Haq infront with a sharp incoming delivery and had the not out decision by New Zealand umpire Billy Bowden overturned by the Decision Review System to boost England's cause.
Haq had put on a valuable 52 runs for the sixth wicket with wicket-keeper batsman Adnan Akmal, who was unbeaten on a gritty 24.
James Anderson ended the day on a high note for his team when he bowled Abdul Rehman (four) in the last over to leave Pakistan at 288-7 at the close, a lead of 96 over England's first innings total of 192.
Haq's wicket has seemingly brought England back in the match. The Pakistani skipper had hit five boundaries during his 154-ball knock.
Before Haq, Pakistan lost the crucial wicket of Younis Khan (37) on the stroke of tea when the experienced batsman offered no stroke to part-timer Jonathan Trott's incoming delivery and was adjudged leg-before.
England took the second new ball and Anderson's first over with it brought the downfall of Asad Shafiq (16) to leave Pakistan at 231-5, in danger of being dismissed for a small lead.
It was paceman Stuart Broad who gave England the breakthrough in the morning session after the Pakistani openers, resuming at 42-0, played out the first 90 minutes without being separated.
Broad bowled Umar with a beautiful delivery which held its line and hit the off-stump, just one ball after Chris Tremlett dropped Hafeez off a miscued hook which resulted in a single.
Umar hit ten boundaries in his 58 and was sedate during his 113-ball stay at the crease.
Three overs later, Broad swung one away from Azhar Ali and had him caught behind by wicket-keeper Matt Prior, much to the jubilation of his team-mates. Ali made just one.
England, who were poor against off-spinner Saeed Ajmal (7-55) on day one, were well short of their best in fielding.
Ian Bell failed to hit the stumps when Hafeez, batting on 44, ran for a single after pushing Tremlett to square-leg and was well short of his crease after being sent back by Umar.
Pietersen also dropped Adnan on nine off Tremlett to cap another tough day for the world's number one Test team, who have only won on two occasions after conceding the lead in a Test in the sub-continent, both against Pakistan in 1961 and 2000.
It was Umar who reached his fifty first, hitting Swann for two consecutive boundaries to complete his 13th Test half-century. Hafeez reached his seventh with a single off Tremlett.
Swann finally dismissed Hafeez with a full pitched delivery as the batsmen missed a sweep. Hafeez hit eleven boundaries and a six during his solid 164-ball knock.
Swann, who took 2-89, failed to trouble the batsmen like Ajmal did on Tuesday but his late wicket brought England back in the match. Broad and Anderson also had two wickets apiece.
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ajmals brilliant bowling should do the trick, It will be better for spinners because the pitch has matured over the past couple of days. Pakistan still firm favourites. Still in the driving seat.