Brown (1-2) still had a chance on 4th-and-goal from the 3-yard line, but after Michael Dougherty completed a pass to Colin Cloherty, Rams safety Adrian Owen knocked Cloherty out of bounds at the 1-yard line to preserve the victory.
Cassidy accounted for 235 yards of offense, passing for 116 yards and rushing for 119 yards and two touchdowns to lift Rhode Island (1-4) to its first win.
Brown took a 35-27 lead with 5:07 left in regulation on Jonathan Edwards' 7-yard run.
The Rams tied it with 1:01 to play in regulation when Cassidy scored on a 1-yard run to cap a 13-play, 70-yard drive, then passed to Leonard for the 2-point conversion.
The teams then traded touchdowns in the first overtime.
Dougherty completed 27-of-47 passes for 407 yards and two touchdowns. Brown receiver Paul Raymond caught six passes for 221 yards, including an 80-yard TD catch.