"Buffalo had to travel east to west, and I thought we could have jumped on them," Coyotes coach Rick Tocchet said. "We've got to score goals. To win in this league, you've got to score. I'm worried about the team in this situation … good teams know they're going to win, and some of our guys were a little sluggish early."
Raanta made 20 saves, but the Coyotes (1-3-0) couldn't follow up on a 3-2 shootout win at the Anaheim Ducks on Wednesday and haven't won two straight games in October since 2015.
Arizona, which outshot Buffalo 36-23, had at least 30 shots on goal in each game but has been shut out in three of four games to start the season, including two at home.
"We have a lot of shots, but I don't think we have those second and third (shots), we don't get rebounds like that," Raanta said. "Even though their goalie played a great game, we are making the goalies look really good right now. I think we have to get more of that eye of the tiger in front of the net and put those pucks in."
Ullmark, making his 25th NHL start and first of the season, made several difficult saves in the second period, when most of the Coyotes' nine shots were high-quality attempts from close to the net.
"It's always fun to make saves, especially if you can make enough saves for the team to win," Ullmark said. "They had some pushes, but we played real solid defensively. We were keeping the guys outside."