Miguel Herrera, the fourth coach in six weeks tasked with guiding the floundering side, used seven players from his own Club America.
The Mexican goals were scored by Rafael Marquez in the 12th minute, Carlos Pena in the 24th, Oribe Peralta in the 47th and Jesus Alonso in the 65th.
Jarkko Hurme scored in the 28th minute for the Finns and an own goal by Juan Carlos Valenzuela in the 56th accounted for their other.
Mexico have not missed the World Cup since 1990 and there were high hopes after they won the gold medal at the 2012 London Olympics.
However, they struggled in the six-nation final round of qualifying in North and Central America and the Caribbean (CONCACAF), finishing fourth to go into the last-chance play-off with Oceania's New Zealand for a berth at Brazil 2014.