Danis made 31 saves for his second straight shutout at home, and Bergenheim scored twice in the New York Islanders' 2-0 victory Saturday.
"I feel really good out there right now," Bergenheim said. "We backchecked well tonight, and we also skated well, too."
The Sabres remained tied with Carolina for eighth place in the Eastern Conference.
"We had plenty of chances to score, and we didn't put any in," Sabres forward Derek Roy said. "It's definitely frustrating."
Patrick Lalime made 32 saves for the Sabres.
The Islanders are 5-1-3 in their last nine games at Nassau Coliseum.
"It feels really good here," Danis said.
The Sabres took seven of the first eight shots of the game, but none of them got by Danis, nor did the other two they had in the scoreless first period.
The Islanders got on the board first at 6:13 of the middle period. On a scramble in front of Buffalo's net, Bergenheim came up with the loose puck in the left circle and sent a quick wrist shot between Lalime's pads. It was Bergenheim's 10th goal of the season, and came on New York's first shot of the period.
"There was a scramble in front, and I couldn't get to it," Lalime said.
The Sabres' best chance to tie the game came in the last minute of the period, but Ales Kotalik hit the left post while Buffalo had a 4-on-3 power play.
Buffalo put a good amount of pressure on Danis in the third period, but it was the Islanders who capitalized when Bergenheim banged in a quick pass from Frans Nielsen with 2:02 left in regulation.
"It was two quick plays," Bergenheim said, "and they both went in."
- Islanders captain Bill Guerin skated in pregame warmups, but was not on the bench for the game, leading to speculation he had been traded.
- With defenseman Brendan Witt suspended for five games by the NHL for elbowing Toronto's Niklas Hagman in the head, the Islanders called up Andrew MacDonald from Bridgeport to play his first NHL game.