Michael Kesselring tied it 1-1 at 6:06, scoring a power-play goal with a wrist shot from the top of the left circle.
Nick Schmaltz scored at 8:25 to give Utah a 2-1 lead. He tapped in a touch pass from Keller at the top of the crease.
Keller then pushed it to 3-1 at 10:31, picking up a loose puck in the right circle and beating Binnington with a turnaround wrist shot.
The Blues’ top line of Neighbours, Robert Thomas and Pavel Buchnevich were a combined minus-seven.
“Our line just can't get scored on. Plain and simple, that's it,” Thomas said. “[We need to] make better plays offensively, hold onto pucks. There's a lot we can do better.”
Neighbours cut it to 3-2 at 1:32 of the second period when he stuffed in a rebound in the crease on the power play.
Cooley scored at 10:42 to give Utah a 4-2 lead, taking a pass from Keller and scoring with a wrist shot from the slot.
“There's no better feeling than winning, especially on home ice,” Cooley said. “These fans deserve it, and obviously it felt like it's been a year since we had a win at home, so it's nice. I thought it was a great effort by everyone in here, a full 60 minutes, and that's how we need to play, especially [against] a team like that. It's an important game. It's basically a playoff game, so it's unbelievable to get those two points.”