Mark Scheifele and Nikolaj Ehlers scored for the Jets (44-24-6), who are 0-5-1 during their skid. Connor Hellebuyck made 24 saves.
Winnipeg is six points behind the Colorado Avalanche for second in the Central Division and four points ahead of the Nashville Predators, who hold the second wild card into the Stanley Cup Playoffs from the Western Conference.
"It's disappointing that we had to battle back to get to 2-2 to start with," Winnipeg coach Rick Bowness said. "We're just finding ways to beat ourselves. Had enough offense for a couple periods. Some uncharacteristic odd-man rushes against us, again, just guys trying to do too much or making a bad read. It's one or the other. It's another game where we had a wide-open net to tie it up. That's back-to-back games. That was sitting there, we just missed it. It's disappointing we had to battle back."
Ehlers gave the Jets a 1-0 lead at 1:58 of the second period, scoring in front off a rebound of Sean Monahan's shot.
"It definitely hurts. It's not a good feeling, that's for sure," Scheifele said. "That's how fine this game is. Just little things that matter, especially at this time of the season. They put the pressure on all game long. [Hellebuyck] made some good saves. I wish we pushed a few more past them."
Katchouk tied it 1-1 at 3:39 on a 4-on-1 rush with Erik Brannstrom.
"We're standing still," Bowness said of the sequence that led to Katchouk's goal. "Any time you're standing still when you make those plays, and they're moving, you're going to put yourself in a bad position. Again, it's just making the right read. Trying to make poor percentage plays and then it's a bad read on top of that. It just magnifies it. That's what it looks like."
Greig gave the Senators a 2-1 lead off a turnover in the Winnipeg zone when he deked Hellebuyck with a backhand at 13:25.
"I think it's just the little things that everyone's been doing, and you see the results that we've had," Tkachuk said. "We've got to keep building, keep finishing strong and keep building on our momentum."