Erik Karlsson and Timo Meier scored, and Josef Korenar made 29 saves for the Sharks (21-27-6), who couldn't hold a 2-0 lead.
"They got behind us a little bit," Sharks defenseman Mario Ferraro said. "He [Kessel] had a couple of breakaway chances and we should have been aware of that. We had a couple of bad breakdowns and some bad luck."
Karlsson appeared to tie it 3-3 at 7:39, but Coyotes coach Rick Tocchet successfully challenged the play for offside. Video review confirmed Tomas Hertl preceded the puck into the offensive zone.
Michael Bunting's 10th goal in 20 games this season made it 4-2 at 9:57.
Jenik, making his NHL debut, scored into an empty net at 19:59 for the 5-2 final, the fifth unanswered Arizona goal.
Kessel is the 13th United States-born player and the second active to reach 900 points (Patrick Kane, 1,086 points).
"Phil is a money player," Tocchet said. "Everywhere he's gone his teammates love him. … He's a pretty special player, he's got a few years left in him to get that 1,000 points. But that [900] is a big, big milestone."
Karlsson's snap shot from center point gave the Sharks a 1-0 lead at 3:47 of the first period.
Meier scored on a 2-on-1 rush with Ivan Chekhovich to make it 2-0 at 3:30 of the second. Chekhovich got the assist for his first NHL point in his second NHL game.
Soderstrom, a defenseman playing his third NHL game, cut around Sharks forward Rudolfs Balcers to score on a wrist shot from the right face-off circle to cut it to 2-1 at 4:59.
"I got a good pass from Oliver Ekman-Larsson, he's a skilled player, and I was going to shoot the puck, but I saw the forward was pretty low," Soderstrom said. "I tried to step up and I got a good shot on net."