Dmitrij Jaskin and Vladimir Tarasenko also scored, Kevin Shattenkirk had two assists, and Allen made 40 saves for St. Louis, which got the two points after failing to hold a 3-0 lead in the second period.
"Nothing changes for me. Just try and stay patient, be big and let them make the first move," Allen said. "Sometimes you go in those shootouts and you get seven or eight goals on 11 shots, this one there's no goals on either side. It's just the way she goes."
Schwartz took part in the ninth round of the shootout and was stopped by Hammond; he replaced starter Craig Anderson, who left with a lower-body injury in the second period. Senators coach Dave Cameron said there would be an update on Anderson's injury Wednesday.
"It was different. I don't think I've been part of a shootout that long with no goals," Schwartz said. "Just great goaltending. There wasn't a whole lot. They cut off the angles good, the dekes weren't really open. They did a good job. The ice didn't help, either. Berglund, that play I think was to just shoot it and try and pick a corner and he did a good job."