Nick Ritchie and Jakob Chychrun each had a goal and an assist for the Coyotes (14-35-4). Scott Wedgewood made 33 saves.
"I think they dominated all the game in my opinion. We had really good goaltending keep us in the game," Arizona coach Andre Tourigny said. "You can see it by the number of giveaways we had. When you don't want to get engaged, you force the play with the puck and you turn it over and are under duress."
Hayton tied it 2-2 at 16:40 of the first when he scored a highlight-reel goal. He cut through the neutral zone, toe-dragged Austin Poganski while splitting three defenders and switched from his forehand to his backhand to score inside the right post.
"Ritchie made a great bump-up to me, and I think they were coming off a change and were a little flat-footed," Hayton said. "I was able to get the guy going to the last, then pull it back and come in with some speed."
Ritchie made it 3-2 at 13:56 of the second. It was his first goal for Arizona since being acquired in a trade with the Toronto Maple Leafs on Feb. 19, and his first goal overall since Jan. 8.
Blake Wheeler's wrist shot tied it 3-3 at 3:17 of the third for the first of three consecutive Jets goals.
Connor's go-ahead goal made it 4-3 at 5:34. After Comrie went post-to-post to make a save on Phil Kessel, the Jets went the other way and Svechnikov found Connor open near the right post.
"Our plan was Eric was going to play here today and he gave us the game we expected," Jets coach Dave Lowry said. "He made some critical saves at key times that allowed us to win."
Connor has scored six goals in his past seven games.
"The top players have got to be the top players, we've got to produce," Connor said. "We're not where we want to be in the standings, it's quite a daunting task, but I believe in this group. I think we can get it done."