Archie White, Mark Lambert, Jack Clifford, Luke Wallace and James Chisholm went over for first-half tries and Ben Botica added two conversions as Conor O'Shea's side took a 29-3 lead into the interval, with Filippo Buscema replying with a penalty.
Matthew Hopper crossed 15 minutes into the second half and Botica added a third conversion before the hosts got their only try through flanker Mata Mbanda and Matteo Minozzi converted.
Ireland beat England at Twickenham to secure grand slam
Kyle Sinckler and Charlie Walker then added to Quins' try tally and Botica converted both.
England hold on for narrow win over Wales
England call up scrumhalf Wigglesworth