Negotiations have been ongoing throughout deadline day, with the move made more complex by the fact that Wague was on a long-term loan at Udinese from Spanish side Granada.
The Foxes have loaned the Mali international defender from Udinese with the consent of Granada, having undertaken lengthy talks with both clubs to get the deal over the line.
"I'm delighted to be at Leicester City and I can't wait to get started. It's a wonderful club and I'm excited to have the chance to show what I can do in the Premier League with the champions," the 25-year-old told the club's official website.
"I've always wanted to play in England so I'm looking forward to the challenge and to getting to know my new teammates over the coming weeks. I'll work hard to get my chance and when it comes I hope to show that I will give everything to help the team."
Wague becomes Leicester's second signing of the January transfer window following the arrival of Wilfred Ndidi.