The Irishman was dismissed after 29 minutes by referee Kevin Clancy following a tackle on Hibs midfielder Kevin Thomson.
Television replays of the incident showed the Buddies skipper making contact with former Rangers man Thomson but Saints boss Danny Lennon believes there was not enough to merit a straight red.
He said after the match in Paisley: “Jim’s hurting. He is distraught. He’s a very honest man and said there was no contact.
“We’ll look at the evidence and if there’s any doubt we’ll certainly be appealing.”
But Thomson was sure the referee called it right, saying: “I thought it was a poor tackle from Jim. I got there first and you could hear it back in Edinburgh - he clattered my shin pads. I thought it was quite nasty, to be honest.”
A Scottish Football Association disciplinary panel will meet next week to consider the appeal, meaning Goodwin - who had collected 14 bookings this season before he was sent off - is free to play against Partick Thistle on Friday night in a crunch fixture which could help decide if either side will face a relegation play-off at the end of the season.
Despite playing for more than an hour against Hibs with just 10 men, Saints won 2-0 to move up to joint seventh in the Scottish Premiership table.