Flights at this price start in the middle of November through early December then again from the mid-January through early March. Departure dates will vary, but you'll have the most luck leaving Newark on Tuesday, Thursday, and Saturdays and returning from Rome on Sunday, Wednesday, Fridays.
If using a metasearch site like Google Flights, tickets will often display higher than advertised, but when clicking through to the booking screen, they will drop in price to the listed rate.
As an example below, I found seats departing EWR on Tuesday, January 22 and returning from FCO on Wednesday, January 30, 2018. Other dates are easy to see during the January and February travel window.
On to the downside... these fares book into United's Basic Economy and only include a carry-on bag plus one personal item. Checked bag fees start at $60 one-way, hence the discount on flight price. Even with the $120 tacked on, $443 is a reasonable price for this route.
There are a lot of great deals to other destinations across the Atlantic on our Europe Flight deal page for your perusing pleasure.
If you prefer to stay stateside or have another international destination in mind, we've got you covered with a complete list of flight deals departing Newark (EWR), New York (JFK), and New York (LGA) here.
Need a room in the Eternal City? Here's what I could find via Hotelwatchdog for the sample dates listed.