Nicolo Zaniolo looked to have clinched a late victory by firing into the bottom corner in the 90th minute, but referee Rosario Abisso disallowed it for a foul by Tammy Abraham in the build-up, after consulting the pitchside VAR monitor.
The furious Italy international was shown a second yellow for dissent moments later, and kicked advertising boards at the side of the pitch in anger as he headed towards the tunnel.
Roma are sixth with 39 points and Genoa are 19th with 14 points, four adrift of safety.
Mourinho was looking to equal a league record held by Antonio Conte by clinching a 62nd win in his first 100 matches in the Italian top flight.
But he had to settle for a point against a Genoa side with one win all season, despite Leo Ostigard's sending-off midway through the second half.
Roma could not make their numerical advantage count until Zaniolo's strike, but the 22-year-old did not take the decision to overturn his goal well and was shown a second booking after exchanging words with Abisso.