"Our officers heard the chants being directed at the official and acted quickly to arrest the two, who are both 17."
They said the pair were arrested "on suspicion of a public order offence", adding: "We don’t tolerate any form of hate and it is important hate crime is reported to us."
Saturday was White Ribbon Day which is the start of a 16 day long campaign that focuses on everyone, especially men and boys, on what they can do to change the behaviour and culture that leads to abuse and violence against women and girls.
Birmingham, who are managed by former England star Wayne Rooney, warned supporters ahead of the match on Friday about their behaviour towards match officials.
In a statement, the club highlighted how Ms Welch faced "sexist and misogynistic" chants during her previous match at St Andrew's against Preston.
The club also reminded supporters a five-year ban could be issued for discriminatory or hate speech.
A statement was issued on BCFC.com "encouraging supporters to be respectful towards the match officials" was issued before kick off.
Birmingham City said it was "deeply disappointed that two supporters were overheard by West Midlands Police officers engaging in misogynistic chanting towards today's referee".