Chrissy Teigen and John Legend are welcoming a new member to their family ... sharing a video of the rescue dog they adopted just months after one of their other pups passed away.

The Hollywood super couple announced the adoption on Sunday -- National Puppy Day ... explaining they adopted Dudley from Wags & Walks ... a nonprofit "dedicated to saving dogs from high-kill shelters and placing them in loving homes."

Legend and Teigen shared a clip featuring Dudley who came into a Wags & Walks shelter shortly after the L.A. wildfires in January.

The duo gives Dudley's story from that point on ... including detailing his struggles while recovering from pneumonia.

Dudley's happy and healthy now ... and, Chrissy and John plan to keep him that way by making sure he lives on Kismet -- the dog lifestyle brand the two stars cofounded.

As we mentioned ... Chrissy and John lost their dog back in January -- telling fans their pup Penny had gone to doggy heaven.

Chrissy wrote a heartfelt message to Penny at the time ... saying she'd miss her -- and, they'd all be together again one day.

No dog can replace Penny ... but, it certainly seems like the family's excited to have Dudley now, too!