Rodrigo Kvernadze put Frosinone ahead in the 14th minute at the Ferraris, capitalising on a defensive mix-up before beating Bijlow, after Genoa had already seen an earlier Baldanzi effort ruled out for an offside in the build-up.
Filip Vasquez levelled from close range early in the second half, meeting a corner from Sow, but Genoa were unable to find a winner after Lorenzo Colombo missed a first-half penalty and Vasquez himself was sent off late on for a professional foul, leaving Daniele De Rossi's side to play the closing stages with 10 men.
De Rossi rejected the suggestion that his side lacked character despite a ninth match without a win in Serie A.
"I don't agree. It doesn't seem fair to say that. It's a team that played its match and deserved to be ahead in the first half. We didn't play our usual game but we did the job we had to do today. The bounces of the ball didn't go our way today, that's part of this moment. We have to move forward, but saying the team lacks personality seems very unfair," De Rossi said.
The Genoa coach argued his side's underlying identity would strengthen as recent signings gained more minutes.
"We came up against an opponent that would have caused us problems if we'd played the way we always do.
"In the first half I think we should have scored three goals. We're open to criticism, but we have a clear identity that will grow based on certain players who have arrived recently and have little playing time," he said, backing Colombo despite his missed penalty.
Massimiliano Alvini felt a point away from home represented a fair outcome from an open contest.
"It was an open match. I think in the context of the game, the draw can be a fair result. Of course we had the chance to win it, and that's where we need to improve. When we could have gone for the kill, we need to get better. It's a significant result after six official matches between league and cup," Alvini said.
The Frosinone coach was particularly pleased with his side's resilience in a difficult away environment, and praised the impact of his substitutes.
"I'm very happy with the performance we put in. Coming to Genoa, to this stadium, with this crowd, and putting in a performance while down to 10 men for the first time is a big thing. The substitutions made a difference. We were eager to go for it, to want to win," Alvini said. "It's an important point, significant for the journey we're on. I'm not angry at all."