ANSWER: First Battle of Cabin Creek, July 1-2, 1863.
SOURCE:
http://www.okhistory.org/publications/enc/entry.php?entry=CA001
Hoosier has final say as I see there are a lot of different answers floating around.
Edit - Here's my ruling.
The Oklahoma History website cited as the source supporting the official answer states in regard to the First Battle of Cabin Creek, "At this engagement the men of the First Kansas Colored Infantry held the distinction of being the first African American soldiers to fight alongside white troops.” So this is supported by a legitimate source.
There were numerous other engagements in which African American soldiers participated prior to the First Battle of Cabin Creek. I can't find any sources stating, in so many words, that the African American soldiers fought alongside (or side by side with) white troops at any of these battles. However, in every instance there were white soldiers in the vicinity. I also think Schwallanscher's argument (that, since the African American units were led by white officers, there were always whites fighting in any action where African American units were involved) has a great deal of merit.
Therefore, I am going to give credit for a correct answer to anybody who named the First Battle of Cabin Creek (which took place July 1-2, 1863) or any of the other battles prior to that date in which African American troops were involved. This includes the actions at Island Mound, Township Landing, Port Hudson, and Milliken's Bend.
I am not going to give credit for a correct answer to those who named the battles of Grimball's Landing, Honey Springs, or Fort Wagner. All of them took place after the First Battle of Cabin Creek, so they cannot claim to be the first battle in which African Americans and whites fought together.
Hoosier