In another heart-melting image the pup is seen sleeping next to the actress
The cute pooch is also seen grinning as he wears a blue cone-shaped party hat
One wrote: "Chesterfield! Love you sweet boy! I can't believe it's been a year!!!"
Another added: "Please drop this handsome man off at my house. He is an ideal date thankyouuuu."
A third commented: "The best duo and the best mama ever, I know lord is so lucky to have you spoiling him on his special day and ALWAYS!"
Hollywood actress Gwyneth Paltrow added: "What a guy."
Jennifer first introduced her pet to her fans on Instagram last October.
In a clip the star could be seen creeping up to her new pet as he snoozed with a bone in his mouth.
She wrote: "Chesterfield, have you fallen asleep with a bone in your mouth? I think you have"
Jennifer then explained in the caption: "Hi! I’d like to introduce to you the newest member of our family…. this is (a very tired) Lord Chesterfield. He stole my heart immediately.
Jennifer first introduced her pet to her fans on Instagram last October
The actress, 52, paid tribute to her pet in the caption
"A HUGE thank you to @wagmorpets for the incredible work you do. Grateful you take such great care of these rescues and find them their forever homes."
Celeb pals and fans alike rushed to post comments underneath the adorable post.
Fellow actress Lily Collins wrote: "I actually cannot deal with this cuteness. The name, the pose, everything. Congrats!!!!"
Millie Bobby Brown commented an emoji with the heart-shaped eyes.
Another fan wrote: "He is so cute! i’m so glad for everything you do to help everyone.
"You are amazing, i’m proud to look up to you! also, welcome to the family."
The Morning Show star has two more dogs – Clyde, a Schnauzer mix, and Sophie, a white pitbull.