Mane was fortunate not to get a red card shortly after that following a poor challenge, but Senegal could not add to their lead and paid the price when the Ivorians were awarded a clear penalty and equalised through Franck Kessie four minutes from the end of the 90 minutes.
Ivory Coast will face the winner of Tuesday's last-16 tie between Mali and Burkina Faso in the quarterfinals on Saturday.
The Ivorians were in disarray just days ago. Hammered 4-0 in a record loss on home soil by Equatorial Guinea, they were on the verge of elimination and sacked coach Jean-Louis Gasset, replacing him with assistant Emerse Fae.
But a fortuitous set of results elsewhere saw them squeeze into the knockout round and they put up a much-improved display against a Senegal side that had won all three of their pool matches with relative ease.
"It's very emotional, the last few days were tough for us," Ivory Coast striker Sebastien Haller said. "We have opened the door and given ourselves a chance. We can forget now about what happened in the group stages."
The hosts were the better side on the night and might have wrapped things up before the need for penalties, but will simply be relieved to fight another day having avoided an embarrassing early exit from their own tournament.
Leading 1-0, Senegal thought they should have had a penalty in the second half when Ismaila Sarr went down in the box, but the match officials decided there was not enough contact to send him sprawling, much to the annoyance of coach Aliou Cisse. Mane also fired wide when he should have scored.
The Ivorians could have been level when Nicolas Pepe had a clear shot of goal but could not beat Edouard Mendy, before they were awarded a clear spot kick on VAR review as the former was clattered in the box by the latter and Kessie buried his kick.
The game went to penalties and the unfortunate Niakhate missed his side's third kick as he struck the post while the Ivorians held their nerve to secure a place in the quarterfinals.