Costa was part of a group of three riders at the front and managed to hold off German Lenard Kamna and Colombian Santiago Buitrago to the finish line in the mountainous 158.5-kilometre stage from Pamplona to Liconbery.
Costa and Buitrago broke away from a group of 15 riders but Kamna caught them and took the lead before falling and handing the advantage back to his pursuers. The German, winner of the ninth stage, recovered and ran quickly to the goal line, but he could not resist Costa.
Kos will wear the leader’s red jersey for the seventh consecutive day when competition resumes on Tuesday, after a day of rest.
The American racer finished the main race to maintain his lead of 1 minute and 37 seconds over his teammate Primoz Roglic of the Jumbo-Visma team.
Kuss is the first American to drive a grand tour since Chris Horner won the Spanish Tour in 2013.
He arrived at the race as a co-rider in the mountains, but should now be considered a real contender for the podium in Madrid on September 17.