Paul's wrist shot from the right hash marks gave the Senators a 1-0 lead at 11:50 of the first period.
Scheifele got to a loose puck in front of the net and scored a power-play goal to tie it 1-1 at 16:35.
Scheifele tipped a Pionk point shot on the power play for his second goal to make it 2-1 at 17:46.
"We knew what they were going to do and we capitalized when we had the chance," Scheifele said of scouting the Senators penalty kill.
Connor scored at the side of the net to extend the lead to 3-1 at 4:52 of the second period. It was Connor's 30th goal of the season; he has scored at least that many for three straight seasons.
"We don't feel Kyle to be very lucky this year. There's more there," Maurice said. "This guy's going to score an awful lot of goals."
Nikolaj Ehlers made it 4-1 off a rebound at 5:54.
Scheifele completed the hat trick when his wrist shot went in off Ottawa defenseman Andreas Englund with 1:33 left in the third period for the 5-1 final. It was Scheifele's third NHL hat trick.
"We just didn't play smart enough tonight," Senators coach D.J. Smith said. "That's really what it is. We turned a lot of pucks over. I didn't think we got to their net nearly enough. We've got to be a little harder offensively."
Defenseman Dylan DeMelo, who was traded to the Jets by the Senators on Tuesday, blocked three shots and played 18:36 in his Winnipeg debut.
"Coaches love those kinds of games," Maurice said of DeMelo's performance. "It's not flashy, but his consistency with his reads, and then his passes … that was a real complete, kind of quiet game that coaches love."
Senators forward Vladislav Namestnikov was scratched for precautionary reasons. The NHL Trade Deadline is Monday.
"Certainly, some of the guys were a little thrown off," Smith said of Namestnikov being held out of the game. "It threw our lines off, but it's part of hockey. It is what it is. Everyone knows this time of year, when you're in our situation, this happens. And you've got to be pros and you've got to get back to work."
