Lambert and Rodriguez scored 52 goals between them last season and were named as the strikers in the PFA's Championship team of the year.
And Lambert told the Saints website: "When I played against him last year I felt he was a very good player. He's a good, young English talent and I'm very much looking forward to playing alongside him."
Cork, who played alongside Rodriguez at Turf Moor, said: "I've heard that it was a lot of money but I'm sure Jay will justify the price tag because I know he's a great player.
"He's quick, he's strong, a good header of the ball, and he can score a variety of goals - not just inside the box, but a few outside as well."