Mike Woodson is super pumped to see the New York Knicks win it all ... telling TMZ Sports it's not a bittersweet feeling, 'cause "once a Knick, always a Knick."

Woodson -- who played for AND coached the Knicks once upon a time -- said it was "unbelievable" to watch Jalen Brunson, Karl-Anthony Towns, and the rest of the squad handle business against the San Antonio Spurs ... especially after decades of failing to get over the hump.

The one guy Woodson made sure to shoutout was Knicks owner James Dolan ... adding he's "truly, truly happy" to see him add a Larry O'Brien in his trophy case.

But it's not just the ownership, front office, and players -- Woodson knows the impact the Knicks' NBA Finals victory has on the whole city ... and that was clear when the team paraded down Broadway last week.

So even though he tried to be the guy to turn the franchise around when he took over as head coach in March 2012 after Mike D'Antoni resigned, there's nothing but positivity.