After falling behind early on against their Russian rivals, it looked like Rafael Benitez's side were on their way to a second successive defeat following Sunday's loss to Chievo.
However, Higuain recovered from missing a penalty in that match to score from the spot before the interval, and Dries Mertens added a brace to help Napoli to three points.
"After Sunday's defeat we had to prove to our fans that we wanted to win and that we wanted these three points," the Argentine forward told Mediaset Premium. "It was an important game, we had to show our character.
"Last year we played in a lot of big games late in the season and this year we want to repeat that. I shot my penalty in the same direction as I did on Sunday - I was not thinking about another mistake but only to score."
Napoli return to action against Udinese on Sunday.