The Gunners made a brisk start on Merseyside despite their Champions League exit mid-week after Danny Welbeck made the breakthrough in the seventh minute.
Arsenal edged into a comfortable 2-0 lead thanks to Alex Iwobi’s first Premier League goal on his English top-flight debut.
Romelu Lukaku, who has scored 17 Premier League goals so far this season, was unable to trouble Arsenal goalkeeper David Ospina.
And former Manchester United midfielder Hargreaves heaped praise on Arsene Wenger’s players for a mature performance at Goodison Park.
“Absolutely. It was a powerful and mature performance,” Hargreaves told BT Sport.
“Iwobi played incredibly well coming into the team as a 19-year-old. Ozil pulled the strings. Arsenal had pace in wide areas: Iwobi, Sanchez and Welbeck. Coquelin and Elneny were brilliant.”
Hargreaves added: “Arsenal were exceptional in all areas.”
Arsenal are in third place and eight points behind leaders Leicester City, who have played one game more than the Gunners.
Wenger’s side are in action against London rivals Watford in their next Premier League game at the Emirates Stadium.
Arsenal haven’t won the Premier League title since 2004.
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