Lamela (30, 61) was joined on the scoresheet by Pablo Osvaldo (23) and Nicolas Burdisso (13) as the home side moved 4-0 ahead against a Milan team toiling to deal with any cross balls into the area.
Substitutes Giampoalo Pazzini (87, pen) and Bojan Krkic (88) made the match sound closer than it was after Marquinhos had been harshly end off with 13 minutes or so left after his deliberate handball was deemed to have denied Stephan El Shaarawy a goalscoring opportunity.
In truth, Milan were awful as they slumped to their first defeat in this fixture since 2008.
Roma have scored the most headed goals in Serie A this term with 10 while AC Milan have conceded the most nine. That was the theme of the night as Roma subjected Philippe Mexes and Mario Yepesto a tortuous night at the heart of the Milan defence.
Milan had won their previous four matches heading into this one, but the pressure is bound to intensify on coach Massimiliano Allegri due to the abject nature of this defeat.
Milan's night of woe began when Burdisso roses to power a header into the net from a Miralem Pjanic corner.
Osvaldo added the second goal from Francesco Totti's cross when he converted with a fine header from five yards out.
Lamela strode onto a ball from Daniele de Rossi before slamming the shot into the net for number three with the Milan defence nowhere to be seen.
Federico Balzaretti helped Roma moved four goals to the good when he picked out Lamela, who rose above Kevin Constant to power a header into the corner of the net.
Erik Lamela has now scored his 10 goals in 14 appearances in Serie A this season
Milan rallied when Marquinos walked as Pazzini stuck a penalty down the ball down the middle after Goicoechea had been booked for dragging Pazzini to the deck having failed to hold a Sulley Muntarishot.
Bojan tucked the ball into the net from close range after Goicoechea had saved from Pazzini's header. It came from an intelligent cross by Stephan El Shaarawy, but there were too few similar moments for Milan during the evening.