After the match Swans boss Kenny Jackett criticised the decision, but appears to have changed his mind after viewing replays of the incident.
Tate will now serve an automatic three match ban.
The Welsh side have been hit by further bad news on the injury front, with captain Garry Monk ruled out for six weeks with a knee ligament injury.
"Garry has badly damaged knee ligaments. He is due to have a MRI scan but it looks as if he'll be out for a minimum of six weeks," Swans physio Richie Evans told the club's official website.
Striker Andy Robinson, who put the Swans ahead on the stroke of half-time at Glanford Park, also faces a fortnight on the sidelines with ankle ligament damage..