Bruce Jenner -- Kids Say Dad Won Over the Nation!

Bruce Jenner hit a world series home run in his Diane Sawyer special ... according to some of his kids.

Sources connected with the family tell us, the boys feel like Bruce won over the hearts and minds of an untold number of people who had negative feelings about their dad's transition.

Speaking of the audience, the estimate is ... 17 million people tuned in.

We're told Kim found the special "very emotional" but was crowing about how proud she was of Bruce and how positive the interview came off.