The actress posted an adorable video Tuesday showing what the couple's morning looked like in NYC after she brought home a German Shepherd pup-aroo ... and then placed it on top of the bed to finish waking her hubby up. Yeah, it's pretty freakin' AWW worthy.

At first, Nick seemed disoriented and confused by what was happening ... but eventually, he realized there was a new member being added to their household.

Looks like the couple has already named their pet -- he goes by Gino, and yes ... he already has his own Instagram account. So far, Gino's been pretty active .. throwing up 3 posts.

Gino's arrival comes just days before Nick and Priyanka's 1-year wedding anniversary. They had several celebrations, if you'll recall, but the one that made it official happened on Dec. 1.