The battling Mehidy led his side to 7-283 at stumps on Wednesday, which represented a comeback from 6-112 during a third day interrupted by rain and bad light.
Nayeem Hasan was also unbeaten on 16.
Mehidy shared a record 138-run stand with debutant Jaker Ali, who made 58 as the hosts wiped out a first-innings deficit of 202 runs.
Bangladesh were bowled out for 106 in their first innings before South Africa replied with 308, thanks to Kyle Verreynne's 114.
Fast bowler Kagiso Rabada (4-35) and left-arm spinner Keshav Maharaj (3-105) then shared seven wickets as South Africa put Bangladesh under even more pressure.
"You just have to believe that (you can win)," Bangladesh spin-bowling coach Mushtaq Ahmed said. "When you have belief and confidence, you can challenge any team. We have to bat as long as we can, score consistently, and get whatever lead we can."
Rabada used the early moisture of the pitch expertly after Bangladesh began the day on 3-101. He pegged back the hosts with two wickets in three balls in the fourth over.
Opener Mahmudul Hasan edged to first slip on 40 after being undone by extra bounce.
Mushfiqur Rahim, who became the first Bangladesh batter to reach 6,000 test runs on the second day, paid the price for a lack of foot movement as Rabada's fuller length delivery skidded through to rattle the stumps when he was on 33.
Maharaj removed Liton Das for seven and South Africa sensed a chance to wrap up an early victory. But Mehidy and Jaker survived a probing spell from Maharaj and Rabada.
Mehidy hit Dane Piedt for a six over his head and drove Maharaj to deep mid-off for a single to raise his ninth Test half-century from 94 balls.
Maharaj finally broke through by trapping Jaker lbw to end their partnership, which was Bangladesh's highest against South Africa for any wicket.
Mehidy, who hit nine fours and one six in 171 balls, continued to frustrate South Africa with support from Nayeem before bad light brought a premature end to the day's play.
Maharaj said: "I think Mehidy and Jaker played really well. We threw everything at them and I don't think it was a case of us taking our foot off the gas."
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