The 2nd baseman was jettisoned from the Rangers to New York last week ... and, in complying with the Yanks' no facial hair policy, he busted out the razor for the first time in YEARS.

And, check out the 27-year-old's new appearance ... he's damn-near unrecognizable -- even to his own 3-year-old daughter!!!

"It was weird. I feel weird,” Odor said of his newly shaved face. "Even my daughter, she didn’t even want to look at me. It is what it is. I’m happy to be here."

Don't worry ... Odor didn't lose any of his baseball magic after he ditched the follicles -- in fact, dude came in CLUTCH for the Pinstripes against the Rays in his 1st appearance with the team Sunday.

After starting the game 0-for-4, Odor knocked in the go-ahead run in the 10th inning with a single to centerfield. He later came around to score, too, helping NYY win 8 to 4.

Good luck in your new smooth-faced journey Rougned ... try not to punch anyone!