Juve have now not tasted defeat in 45 league matches, and went ahead through Andre Pirlo early on.
Further goals from Arturo Vidal and Alessandro Matri put the game beyond Roma, who responded through Pablo Osvaldo, but Sebastien Giovinco then netted a fourth late on for Juve.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.