Mushfiqur struck a superb 137 off 233, smashing 12 fours and one six to help Bangladesh post 390 in the second innings.
Litton Das followed his first-innings 126 with an elegant 69.
Mushfiqur's 14th century saw him surpass Mominul Haque to become country's most prolific Test centurion.
The task Pakistan faces is stark: no team in Test cricket history has won a match chasing more than 418 runs.
But bowling coach Umar Gul said the team were ready to take on the challenge.
"To chase 437 runs, you have to be brave and play positive cricket. In cricket, anything is possible, and there is a possibility for both teams," he said.
"We have two days left, we are mentally prepared that there is enough time. If we bat through the full duration, there is a chance to win because we aren't just looking for a draw. We are prepared for the chase."
Pakistan batted two overs on day three without scoring a run before bad light forced an early close.
Khurram Shahzad, who claimed 4-86, trapped skipper Najmul Hossain Shanto lbw with a sharp inswinger in the fourth over for Pakistan's only success in the morning session after the hosts resumed at 3-110, building on their 46-run first innings lead.
Das and Mushfiqur frustrated the Pakistan attack, adding 123 runs for the fifth wicket.
Pacer Hasan Ali undid Das with a short-pitched delivery to break the stand and Shahzad removed Mehidy Hasan Miraz (19) to give Pakistan a glimmer of hope. But Mushfiqur found another reliable partner in Taijul Islam (22) to keep Pakistan bowlers frustrated.
They added 77, Mushfiqur then guided the lower order astutely to take the lead well past 400 before being last out to Sajid (3-126).
"From what we have seen that the wicket is still good," Taijul said.
"But the target we set gives us psychological edge. When the opposition sees such a target, many thoughts come into their minds. Still we have to stay disciplined."
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