With a 106-run lead from the first innings, Bangladesh set Afghanistan a target of 479 runs for victory.
Bangladesh resumed the day at 223 for six in their second innings. Litton and Mehidy then put together a strong partnership, adding 114 runs without losing a wicket in the morning session to take Bangladesh to 337 for six at lunch.
The pair continued their fluent batting after the break, with Litton bringing up his seventh Test century. The wicketkeeper-batter reached the milestone off 115 balls.
Earlier in the innings, Litton also became the fifth Bangladesh batter to score 9,000 runs in international cricket, joining Mushfiqur Rahim, Tamim Iqbal, Shakib Al Hasan and Mahmudullah.
At the other end, Mehidy brought up his half-century off 92 balls, providing valuable support as Bangladesh pushed their lead towards the 500-run mark.
However, Litton was dismissed after reaching his century, prompting Bangladesh to declare at 372 for seven.
Afghanistan now need 479 runs to win the Test, while Bangladesh must take all 10 remaining wickets to secure victory.
