The former Burnley man has signed a four-year deal with the Saints worth a reported club-record £7 million and his arrival has been met with approval from his new teammates
Rodriguez has signed a four year deal with the Premier League new boys worth a reported £7m, eclipsing the club's former record purchase of £4 million for Rory Delap.
The 22-year-old was named alongside Lambert in the PFA Championship Team of the Year, and his potential future strike partner believes the former Burnley man will prove a shrewd piece of business from Nigel Atkins.
"When I played against him last year, I felt he was a very good player," Lambert told Southampton's official website.
"His team didn't do so well down here when we played them, but I still think he had a good game, and he had a very good game up at their place against us, so I think he's a good signing for us.
"He's a good, young English talent and I'm very much looking forward to playing alongside him."
Cork believes the striker will offer the team a new dimension and hopes he can carry his impressive form from his time at Burnley into the Premier League.
"'Jay Rod' is a striker who's got a bit of everything really," Cork added.
"What I saw of him when I played at Burnley is that he's quick, he's strong, a good header of the ball, and he can score a variety of goals as well not just inside the box, but a few outside as well.
"Like all of the lads he'll be trying to break into the side, so I think pre-season training will be as sharp as anything because everyone will want to prove that they can do it at this level.
"It's one of the best leagues in the world and everyone wants to show that they can play there.
"Hopefully he can replicate his form here and get some goals in the Premier League which would be good for him and the club too."