Full-back Shaw took to social networking site Instagram to praise Lambert's contribution to Southampton's rise from League One to eighth in the Premier League.
|6/1||Southampton are 6/1 with William Hill to be relegated in the 2014-15 season|
And the 18-year-old revealed he hopes his fellow England World Cup squad member can continue his fine form of recent years at Anfield.
"So happy for Lambo, he deserves everything he gets. Been a massive part of Southampton's recent success, his dream really is coming true and I wish him all the best and success in his new journey. Thanks for everything," he wrote alongside a picture of he and the 32-year-old celebrating during the 2013-14 season.
Shaw wasn't the only member of the Southampton squad to show his gratitude to Lambert, with defender Jose Fonte also wishing him the best of luck on what the striker has described as a dream move.
"Mixed emotions!!! So sad to see you go but also happy you achieved one of your dreams!! We had many great moments together that I will never forget!! We got where we got mostly because all your goals!! Thanks for everything and I know you will keep scoring and stepping it up!!" he wrote on his Instagram.
Lambert is the first player to leave St Mary's this summer, though with Adam Lallana also interesting Brendan Rodgers and Shaw on Manchester United's radar, he may not be the last.