The Poland Under-21s international, who was a regular for Lech last season, has been handed a five-year deal at St Mary's Stadium.
Southampton have not revealed the sum of money handed over to the Polish club, but they are said to have had a £4.5m bid knocked back earlier in the window.
Bednarek is delighted to have got a deal over the line and is relishing the chance to make his mark in the Premier League with his new side.
"I am so happy that I could join a club like Southampton," he told the club's official website. "This is the moment I have worked for that makes my dreams come true. It only gives me motivation to work hard, to show my skills and to help Southampton to get better and better.
"I think it's a good step for me, because I have heard Southampton is such a good club for young players, so I can improve here. I think it will be a good move for me."
Vice-chairman Les Reed added: "Jan is an extremely promising young player who possesses significant potential. We are delighted to bring him to the club, and we believe this will be the ideal place for him to continue his development.
"This is a particularly exciting time for him to arrive here, as he also has the opportunity to learn from Mauricio Pellegrino's extensive experience of playing the centre-back position.
"Jan provides additional depth to the impressive amount we have in that area, with Virgil van Dijk, Maya Yoshida and Jack Stephens having all performed so impressively there last season, and he provides us with another excellent central defensive option."