The right-handed batsman slammed 143 to make an impact.
Babar has moved into the top five of the ICC Test Rankings for Batsmen for the first time in his career.
Babar rises to fifth, Shafiq and Masood gain
Babar went past David Warner and Cheteshwar Pujara and is now placed fifth in the ICC Test Rankings.
He has 800 rating points and is 14 behind New Zealand's Kane Williamson (814).
Meanwhile, Asad Shafiq, who slammed 65, has moved up to 18th in the rankings.
Centurion Shan Masood also rose to 32nd in the rankings and has 582 rating points.
Records scripted by Babar against Bangladesh
Babar slammed his fifth career Test century.
This was the fourth century for Babar in his last five Tests and sixth fifty-plus score in last seven innings.
The right-handed batsman raced to 1,850 career Test runs.
He went past former Pakistani great Shahid Afridi (1,716) in terms of Test runs.
Babar also registered his highest Test score.
He slammed 19 fours and a six.
Pacers Abbas and Afridi gain as well
In the ICC Test Rankings for Bowlers, pacer Mohammad Abbas has risen to 14th. He now has 751 rating points after surpassing Nathan Lyon and Tim Southee. Shaheen Afridi, who claimed five wickets in the match, rose to 32nd and has 481 rating points.
World Test Championship: Pakistan have 140 points
Pakistan are stationed at fourth in the ICC World Test Championship. They picked up 60 points after the win and moved ahead of Sri Lanka. Pakistan have collected 140 points after five Tests. They have won two, lost two and drawn one.