Philipp Kurashev gave Chicago a 1-0 lead at 16:22 of the first period, picking up a pass from Jarred Tinordi in the neutral zone and moving down the left wing before scoring from the left face-off circle.
Jones made it 2-0 at 2:00 of the second on a give-and-go with Bjork, then extended the lead to 3-0 at 5:29 when he took a cross-ice pass from Bjork and scored on a wrist shot from the left circle.
Bjork was acquired in a trade from the Buffalo Sabres last Thursday.
"Yeah, it felt great," he said. "Those were two great shots that he had there, so [I'm] glad I passed to him. Those were unbelievable goals. That was a great win for us.
"I thought we played the right way the whole game and everyone played well. My communication with my line (with Kurashev and Lukas Reichel), we were talking about playing the right way, and I think we did a really good job of that and tried to do that all three periods, and we were pretty successful."
Jason Dickinson made it 4-0 from the slot at 10:31 on a pass by Bjork from behind the net.
Ottawa outshot Chicago 27-14 through the first two periods.
"Right from the get-go, just attention to detail, odd-man rushes that we gave up … it was everything that we hadn't done in the previous five games, we did tonight," Senators coach D.J. Smith said. "It's tough to explain how well we've played to come out and do this. But you've got to turn the page quickly.
"We were feeling so good playing at home, and then to come out here -- just no rhythm, no feel, team wasn't prepared. That's on me. My job is to get them prepared and to push them every day, and we weren't ready to play today."