The Islanders were shut out for a second straight night. They were blanked in a 3-0 loss to the New Jersey Devils on Saturday at Barclays Center in Brooklyn.
Anders Nilsson stopped 23 of 25 shots for New York.
"The one thing for me is that we've gone six periods of hockey without a goal," Islanders coach Jack Capuano said. "So that tells us about some of the things we need to work on as the week moves forward."
Bourque broke a scoreless tie 3:07 into the second period, when he beat Nilsson with a wrist shot. Watson then made it 2-0 with a wrist shot of his own at 8:22.
"I think I've tried to be consistently good every day," said Watson, Nashville's first-round pick (No. 18) in 2010. "I'm just coming out here, working hard and just trying to do the good things in my game that are going to hopefully help me keep around."
Rinne played the first 40 minutes for Nashville and made 13 saves. Hutton stopped all 12 shots he faced in the third period.