Referee Peter Walton told the FA he would have sent off Bowyer had he seen the incident during City's 3-0 loss to title challengers Arsenal at St Andrew's on Saturday.
The 34-year-old will now miss Tuesday's league match at Blackpool, the FA Cup third round trip to Millwall on Saturday and next Tuesday's League Cup semi-final, first leg at West Ham.
Television replays showed Bowyer standing on Sagna when the defender was on the ground near the touchline.
"We accept the decision and will now have to move forward into the next three games without the services of Bowyer," Birmingham manager Alex McLeish told the club's official website. "We have got a good discipline record at this club and stress to the players every week how important discipline is."
Bowyer will be available for the derby clash at home to Aston Villa on 16 January.
Arsenal are two points behind league leaders Manchester United while Birmingham are currently second bottom, with only basement club Wolverhampton Wanderers below them in the table.