The club's worst fears were realised last week when scans revealed the full extent of the centre-half's injury, sustained during the 1-1 draw at Middlesbrough at the end of last month.
"Anthony will see a specialist in London and have an operation on the Achilles," manager Dave Jones told the club's official website.
"It's a blow to the boy and the team because he was playing very well and had a good start to the season.
"We wish him all the very best in his recovery and look forward to welcoming him back as soon as possible."
Gardner had been relatively injury-free since arriving on a free transfer after his release from Crystal Palace in the summer of 2012 and had started in all the Owls' matches this season.
Miguel Llera and Martin Taylor will be vying to partner summer signing Kamil Zayatte at the heart of defence against Yeovil on Saturday.