The Cavaliers (2-0) set a school record from the free-throw line, going 19 of 19, while Winthrop (0-2) made just 9 of 19 foul shots in the Paradise Jam.
The tournament games now shift to the U.S. Virgin Islands, starting Friday. Virginia plays TCU and Winthrop takes on Marquette.
"In the second half I thought we started attacking the ball screen better and we imposed our will defensively and kind out of outlasted them," said Virginia coach Tony Bennett.
The Eagles took a 44-41 lead after the second of back-to-back Andre Smith 3-pointers before Virginia answered with a 12-0 run to go up 53-44. The spurt was highlighted by a KT Harrell fast-break layup that was set up by a Joe Harris steal, and then a Harris 3-pointer.
A Reggie Middleton circus shot stopped the bleeding, but Virginia responded with a 16-2 run, which included an Evans' 3-pointer and fastbreak layup off a steal, to close the game.
"It was fun scoring 17, but that wasn't my focus," Evans said. "I just wanted to come out there and play a good game and give my team a lift."
Virginia played its second straight game without senior guard Sammy Zeglinski, who continued to nurse a sprained ankle.
"You know, I'm disappointed but very proud of my team," said Winthrop coach Randy Peele. "The bottom line is that with 10 minutes to go in the game I think the score was tied and we scored four points in the last 10 minutes of the game."