Jennifer Lopez is making a big relationship move with A-Rod ... introducing him to mom.

The new couple went out Sunday in NYC hand-in-hand -- with J Lo's mother, Guadalupe Rodriguez (yes, really), tagging along for the sight-seeing. 

A-Rod seemed to leave a good impression on mama bear -- we know she's gotta like his last name -- and don't forget ... Jen already hit it off with Alex's sis.

It's pretty obvious now -- he has J Lo's love.

STEPPING OUT IN THE CITY
AKM-GSI