A second-half goal from the spot by substitute Sadick Adams, on his debut, cancelled David Agbordome's opener for the visitors.
Kotoko had won their last two games under coach Steve Polack, who replaced Frimpong Manso, while Olympics had seven points from their previous three games.
Yakubu Mohammed had an opportunity to open the scoring by the fifth minute but he was unable to beat Olympics goalkeeper Herod Quafio, with Baba Mahama also shooting wide three minutes later.
Emmanuel Asante, who is on loan from Kotoko to Olympics combined with Coffie Bekoe on 35th minute but the latter couldn't send his effort on target.
On 48th minute, substitute Kwame Boateng dribbled two Kotoko defenders before releasing Agbordome, who fired past goalkeeper Ernest Sowah.
Kotoko reacted with a substitution as striker Sadick Adams came on for Yakubu Mohammed and the former Atletico Madrid youth player almost got the equaliser with his first shot, which narrowly missed the target.
Two minutes later, Sadick won a free-kick on the edge of Olympics' box but his resulting effort was parried to corner-kick by Quafio.
Kotoko made double substitutions on 70th minute, with Abass Mohammed and Sarfo Gyamfi replacing Obed Owusu and Baba Mahama respectively and they finally found the breach in Olympics' defence seven minutes later.
Olympics defender handled the ball in the box and Sadick equalised from the spot.
The hosts were reduced to ten men on 85th minute after goalkeeper Sowah denied Bekoe an obvious goalscoring opportunity. With the Kumasi-giants having completed their substitutions, striker Abass turned goalkeeper but Olympics failed to make their numerical advantage count.
Kotoko sit fifth position with 25 points, 10 behind Wafa while Olympics have moved one place up to 12th position with 19 points.