Brad won Best Supporting Actor at the National Board of Review Annual Awards Gala on Wednesday night, and after being handed the award and hugging it out, he praised Bradley in his speech for being a great dad ... and a great friend.

Pitt said Cooper rushed to the event after putting his daughter to bed, called him a sweetheart, then added ... "I got sober because of this guy, and every day has been happier ever since."

As you'll recall ... Brad's alcoholism was a major bone of contention in his marriage with Angelina Jolie, and culminated when he allegedly got wasted on a private jet flight and had an altercation with their eldest son, Maddox.

Jolie filed for divorce just days after the incident, and there was a subsequent child abuse investigation ... though Brad was eventually cleared.

Following his split with Jolie, Brad spent over a year in Alcoholics Anonymous and has been sober ever since ... but until now, Cooper's support in the process was unknown.

BTW -- Pitt's victory speech wasn't all about sentimental thank yous ... he also had some good jokes, including one at the expense of George Clooney.