Here's an article about Ten Broeck at the web site of at The National Museum of Racing and Hall of Fame. It makes it pretty clear that the race must have happened in 1878.

And here's a page about African-American jockeys, trainers, etc., at the Kentucky Oaks web site. It has a paragraph about William Walker, an ex-slave who rode Ten Broeck at the famous race.

There's another article at the California Bluegrass Association.