Needing 143 more runs on the final day with 10 wickets in hand, Bangladesh knocked them off for the loss of four wickets when Shakib Al Hasan hit the winning boundary.
Bangladesh recorded only their second bilateral series win of two or more Tests away from home - the last was against the West Indies in 2009.
The tourists won the opening Test by 10 wickets and required just four days in the second after the first day was washed out in Rawalpindi.
"It (the series win) means a lot ... really happy," Bangladesh captain Najmul Hossain Shanto said. "Everyone chipped in in this series, it's totally a team game. I hope this culture will go forward."
Coming into the series, Bangladesh had lost 12 of their last 13 Tests against Pakistan and drew the other in 2015 at home.
But Shanto said the team believed in themselves this time.
"Before we came here we were looking to win," Shanto said. "The way everyone did their job, really happy ... the most important thing is their work ethic."
Bangladesh were 6-26 in the first innings but Litton Das's 138 and Mehidy Hasan Miraz's 78 allowed Pakistan only a 12-run lead.
Hasan Mahmud, Nahid Rana and Taskin Ahmed then dismissed Pakistan for 172 in the second innings - the first time Bangladesh fast bowlers have taken all 10 wickets in an innings - and left Bangladesh a victory target of 184 in four sessions.
The top order showed plenty of resilience to flatten Pakistan plans of a dramatic comeback on the final day, with meaningful contributions from opener Zakir Hasan (40), Shanto (38) and Mominul Haque (34). Shakib, 21 not out and Mushfiqur Rahim, 22 not out, expertly finished the chase at 4-185 in the 56th over.
Pakistan missed chances when they chose not to review when fast bowler Mohammad Ali found the edge of Zakir's bat in the third over on Tuesday after Bangladesh resumed on 42 without loss. And Salman Ali Agha couldn't hold onto a difficult catch of Shadman Islam in the slips.
Shanto and Mominul then kept Pakistan's pacers at bay and watchfully played out leg-spinner Abrar Ahmed, who finally got his first wicket in the match when Mominul miscued to mid-off.
The result extended Pakistan's miserable home record in Tests to 10 consecutive matches without a win in 2 1/2 years. That's happened only once before, when Pakistan lost 11 straight at home from 1969-75.
Shan Masood is the first Pakistan captain to lose his first five Tests.
"Extremely disappointed, especially when you start the home season," Masood said. "The story has been a bit of the same, like, Australia (series lost 3-0 in January). We haven't learnt our lessons."
England come to Pakistan next for three Tests next month.
"It's never doom and gloom," Masood said. "You always keep trying, you always come back. You try and give people a chance and I think that we're heading in the right direction."
The win lifted Bangladesh to fourth in the World Test Championship table, ahead of England.
