England - needing 324 runs to win - were dismissed for 252 late on the fourth day with fast bowler Umar Gul grabbing 4-61 and offspinner Saeed Ajmal 4-67.
Matt Prior delayed the inevitable with an unbeaten 49 off 58 balls before he ran out of partners.
Pakistan won the first Test by 10 wickets and the second by 72 runs.
The whitewash has now put England in danger of losing the ICC's jackpot of $175,000 which is awarded to the top test team until the April 1 cut off date every year.
South Africa can claim the bonus with a 3-0 win over New Zealand in March.
Scores:
Pakistan 99 (Asad Shafiq 45; Stuart Broad 4-36, James Anderson 3-35) and 365 (Azhar Ali 157, Younis Khan 127; Monty Panesar 5-124, Graeme Swann 3-101) def. England 141 (Andrew Strauss 56; Abdur Rehman 5-40, Saeed Ajmal 3-59) and 252 (Alastair Cook 49, Matt Prior 49 not out; Umar Gul 4-61, Saeed Ajmal 4-67)