The 19-year-old became the youngest bowler to take six wickets in the history of the tournament, and gave up just 35 runs in the process - the best figures for a Pakistan bowler in the competition.
"The way Shaheen is bowling, the consistency is amazing. One of the best bowling performances I've ever seen," Pakistan captain Sarfaraz Ahmed said.
Earlier in the day, Pakistan opener Imam-ul-Haq made a run-a-ball century and Babar Azam scored 96 as they set the Bangladeshis a target of 316 runs in both sides' final group game.
After fellow opener Fakhar Zaman was out for 13, Imam and Azam added 157 for the second wicket, laying the foundations for a big total.
Mohammad Saifuddin denied Azam a century when he trapped him lbw, while Imam's innings ended in a hit-wicket dismissal when he backed into his own stumps as he tried to flick away a ball from Mustafizur Rahman.
Mustafizur went on to claim five Pakistan wickets for 75 runs.
In reply, Bangladesh's Shakib Al Hasan scored his seventh half-century of the tournament - equalling Sachin Tendulkar's record of 2003 - before slashing wildly at a ball from Shaheen and being caught behind for 64.
That knock made Shakib the highest-scoring batsman of the World Cup with 606 runs. But Bangladesh's other wickets fell with alarming regularity as they were skittled out in 44 overs.
"I want to say sorry for Shakib. If we stepped up, the tournament could've been different," Bangladesh captain Mashrafe Mortaza told reporters.
Despite the win for Pakistan, the team's poor net run rate meant the 1992 champions had no real chance of pipping New Zealand to the last semi-final slot.
Based on their own score, Pakistan would have had to bowl out Bangladesh for no more than seven runs in order to do that.
"We played good cricket but haven't qualified. That one game against West Indies cost us the tournament," Sarfaraz said, referring to their crushing opening loss.
"We have two months off and we need to do a lot of work with the team," he added.
