Mehidy became the third Bangladeshi player after Shakib Al Hasan and Sohag Gazi, and 39th overall in world cricket, to score a century and take a five-for in the same match.
He struck 104 for his second century as Bangladesh finished their first innings on 444 for a 217-run lead.
Mehidy then took 5-32, his third five-wicket haul in two Tests, as Zimbabwe were were limited to 111 in 46.2 overs in an extended last session on Wednesday's third day.
Left-arm spinner Taijul Islam ended 3-42, after taking 6-60 in the first innings.
The victory earned Bangladesh a series draw after losing the first Test by three wickets.
On an increasingly deteriorating pitch, Bangladesh's two spinners tormented Zimbabwe batters, turning the ball viciously right from the start.
Opener Ben Curran was the only one to offer some resistance, scoring 46 before being the eighth batter out.
Skipper Craig Ervine (25) and Wellington Masakadza (10) were the only others to reach double digits, while three Zimbabwe batters were out for a duck.
Earlier, Mehidy led the lower order astutely after Bangladesh resumed on 7-191 and a 64-run lead.
Taijul Islam and debutant Tanzim Hasan gave Mehidy support as the hosts gained a lead that appeared to be match-winning.
Taijul scored 20 and added a 63-run partnership with Mehidy before Vincent Masekesa got the better of him.
Mehidy then shared a 96-run partnership with Tanzim for the ninth wicket, taking the lead past 200.
Wessley Madhevere broke them up by dismissing Tanzim, who made 41 off 80 balls.
Tanzim's dismissal left Mehidy in limbo for his century as No. 11 batter Hasan Mahmud doesn't have a reputation as a batter.
However he raised second his century by clipping Madhevere past short fine leg for a single.
A tired shot against Vincent Masekesa's slower delivery then cost him his wicket as Bangladesh's innings came to an end.
Masekesa finished with excellent figures of 5-115 on his debut.
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