The on-loan Newcastle man gave his side a 49th-minute lead with a close range finish before Esteban Burgos added a second from the penalty spot five minutes later.
In doing so, Muto became the first Japanese player to score in the top flight in Germany, England and Spain.
The 28-year-old joined Eibar for the season in September and was substituted in the 78th minute of Monday's win.