Per
http://www.history.com/topics/american-civil-war/black-civil-war-soldiers, the first colored troops formed (all before officially authorized by the US Army) were: 1st. 2nd, 3rd Louisiana Native Guard (which fought at Port Hudson, May-June 1863); 1st Kansas Volunteer Infantry (which fought at Island Mound, Oct. 1862); 1st South Carolina, African Descent ( first expedition Nov. 1862). Per
http://www.oac.cdlib.org/findaid/ark:/13030/kt0d5n99qh/entire_text/, the 1st SC did not participate in any actual battles, although they were part of the occupation of Jacksonville, FL, March 10, 1863.
Per Wikipedia article, "Skirmish at Island Mound," and
https://www.kshs.org/kansapedia/first-kansas-colored-infantry/12052, the first Civil War battle in which African-American troops fought was at Island Mound, Missouri, October 29, 1862). Units involved were the First Kansas Volunteer Infantry (Colored) and scouts from the 5th Kansas Cavalry. While the latter contained (per above Wikipedia article) some Cherokee and colored troops, it appears to have been a mostly white regiment. The 1st Kansas Infantry had white officers
https://law.wustl.edu/staff/taylor/preserv/v4n1/battle.htm. However, I can't find a reference that mentions black and white troops fighting side-by-side. Also, most sources call Island Mound a skirmish rather than a battle.
I then looked at Port Hudson, but per this source
http://www.historynet.com/americas-civil-war-louisiana-native-guards.htm, it appears that the 1st and 3rd Louisiana Native Guard were (due to poorly executed battle plans) fighting pretty much on their own.
Since I can't find proof that black and white troops fought side by side at either Island Mound or Port Hudson, although personally I suspect they did to some degree, I have to go with the
First Battle of Cabin Creek, July 1-2, 1863. Per Wikipedia article, "Battle of Cabin Creek," and the Oklahoma Historical Society article,
http://www.okhistory.org/publications/enc/entry.php?entry=ca001:
"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." [quote from the OK Historical Society, but stated in almost identical words by the wikipedia article.]