The Bruins outshot the Lightning 19-5 in the first.
Tampa Bay shuffled its forward lines for the second period, but that didn't stop Boston from extending its lead to 3-0 when Krug scored on a one-timer from below the right dot at 5:59.
Sustr scored his first goal of the season with a wrist shot through traffic at 10:53 to cut the Bruins lead to 3-1.
Although Boston outshot Tampa Bay 13-7 in the second period, the Lightning had a couple of quality chances to get closer.
Rask stopped Tyler Johnson's shot with his mask with 1:29 remaining, and then he pushed off right to left to stop Nikita Kucherov's wrist shot from the top of the right circle on a 3-on-2 with 31.2 seconds remaining.
"We talk about the goaltender needing to make saves and then the team playing well in front of him, and that's where it got away from us," Cassidy said referring to Rask's losing streak. "So that's where he helped himself big time tonight. That's what we need."
Stamkos scored on the power play from the bottom of the left circle to make it 3-2 at 2:10 of the third period, one-timing a pass from Kucherov from the right circle.
Tampa Bay outshot Boston 9-4 in the third period.