Athanasiou made it 2-0 with an empty-net goal with 59 seconds remaining in the third period.
"Those are big moments and big times in the game," Athanasiou said. "Obviously, you want to be out there and be able to help get the puck out, ice the game, and fortunately it worked out tonight."
Brown scored another empty-net goal with seven seconds left for the 3-0 final.
"It kind of goes hand in hand with what I've said all along: Defense is much easier than offense and offense takes longer to develop and evolve," Sabres coach Don Granato said. "And tonight you saw. When a team wants to shut it down, you have to evolve. This happens in every playoff series there is. That's why the statistics are you have a lead after one, lead after two, the winning percentage is astronomical for this reason. If a team is good at shutting it down, offense is extremely hard. For us, we didn't get the win, we didn't get any points, but the video is actually … the ability to learn from this is actually really high and significant."
NOTES: Sabres forward Alex Tuch left the game at 11:05 of the third period after crashing head first into the post. "No head injury, so that's ruled out," Granato said. "He'll just get some imaging. He'll be sore because of the impact, but the testing here was as positive as it could be. Now we'll get some imaging to make sure it's nothing more that could be hidden." … Kings forward Viktor Arvidsson left the game at 4:49 of the third period with a lower-body injury and is day to day, Yawney said. … Los Angeles coach Todd McLellan is expected to be out of COVID-19 protocol and back behind the bench against the Boston Bruins on Monday. … Sabres defenseman Rasmus Dahlin had his eight-game home point streak end. … Buffalo was shut out for the fifth time this season.