Evans started on pole position and beat runner-up Cassidy by 1.6 seconds with Maserati's Maximilian Guenther (26) third in the manufacturer's home race.
The Jaguar driver's fourth career win on the streets of the Italian capital, and third victory of the season, thrusts him more into the title reckoning with three races remaining.
Cassidy now has 171 points, with Andretti driver Jake Dennis (28), the previous overall leader, on 166, and Evans moving up to third on 151.
Only 27 points separate the top four, with Porsche's Pascal Wehrlein (28) on 144 points.
The race was red-flagged on the ninth lap after Jaguar's Sam Bird (36), who started alongside Evans on the front row and took the lead initially, lost control at speed through turn six and spun.
Bird's car was hit by Envision's Sebastien Buemi (38) with DS Peske's Stoffel Vandoorne (31) crashing as he tried to avoid the incident and Maserati's Edoardo Mortara then hitting the Jaguar hard.
Other cars were also involved in the mayhem, with 14 drivers able to re-start after a long break and none of the others injured.
"It was almost two races that we had today," said Evans, the first driver to win in Rome from pole. "Obviously Sam had a pretty scary moment, so good to see everyone is fine.
"A huge result today, maximum points, which was what I needed. Only a small dent into Nick's lead, but it's better than nothing."
McLaren's Jake Hughes (29) was unable to start the race due to a crash in qualifying.
The second race of the double-header is on Sunday.
