Anderson suffered the injury when he stumbled in the build-up to Partick Thistle's goal during the 1-1 draw in the Scottish Premiership on Saturday.
According to the club's website: "Attempts to re-set the thumb on the pitch, in the treatment room and in Perth Royal Infirmary all proved unsuccessful and it took a lengthy operation under general anaesthetic in Dundee on Saturday evening to reset it and repair tissue and bone damage around the thumb."
Saints now expect the player to be sidelined for between six and eight weeks.
Manager Tommy Wright said: "It's a blow for Steven and for us.
"We'll get specialist help to see what can be done to get him back a bit quicker than the current estimates and Ando has a reputation as a quick healer which helps.
"It might leave us a bit short on numbers at the back but certainly not in terms of quality as I have Gary Miller and Dave Mackay who can both move into that role."