Jacob Bryson pulled off a shot and Asplund shaved the return with a backhand for 4-3.
With 24 seconds left, Casey Mittelstadt punctured the match by getting just over the center circle, so as not to risk icing, and put 5-3 in the empty box.
After the lousy suite of 18 straight losses that was broken against Philadelphia on March 31 (win 6-1), there have been four victories in the last seven games for Buffalo who will miss the playoffs for the tenth year in a row.
The 23-year-old forward Asplund came to the NHL 2019 and has played more than 40 games in the league.
Another Swede who impressed for Buffalo was the team's goalkeeper Linus Ullmark.
The 27-year-old saved 40 shots and continues to be one of the few glimmers of light in the team this year.