Adam Fox added a goal and an assist and Alexandar Georgiev made 25 saves for New York, which has won 10 of its last 11.
"We're a confident team," Strome said. "We're coming to the rink expecting to win games."
Patrick Kane and Alex DeBrincat scored for the Blackhawks and Kevin Lankinen finished with 28 saves. Chicago has lost four of its last 11.
"It's a tough loss," Chicago's Seth Jones said.
Georgiev started for the first time since Nov. 21 after replacing Igor Shesterkin in the third period of Friday night's win against San Jose. Georgiev came in 2-1-1 record with a 3.89 goals against average and .867 save percentage this season.
He was sharp throughout, starting with a save on Josiah Slavin 6:18 into the game in which he moved side-to-side to turn away an attempt from the slot. Georgiev also benefitted from good fortune as a Kane wrister hit the post 5 1/2 minutes later.
"It was huge," said Georgiev, when asked if he needed a strong game to boost his confidence.
However, Chicago broke through first on a controversial Kane goal with 22.8 seconds left in the first period. Seth Jones began a rush up the ice before firing a high cross-ice pass that Brandon Hagel knocked down with a backhand. The puck bounced away from Georgiev and Alexis Lafreniere to Kane, who had a half-empty net to deposit his seventh of the season.
Rangers coach Gerard Gallant challenged the goal, presumably arguing the puck was played with a high stick but a review ruled that the puck was below Hagel's shoulder when he played it with his stick.
Strome lasered a power-play goal over a diving Lankinen 4:41 into the second during a four-minute man advantage to tie it.
"We can be better," Panarin said of a power play that entered the game ranked ninth in the NHL in efficiency, and went 1 for 4 against Chicago.
New York took its first lead of the game with about 2 1/2 minutes remaining in the period on Fox's fifth of the season. The reigning Norris Trophy winner snuck to the top of the crease and poked Panarin's centering pass past Borgstrom and Lankinen for the go-ahead goal.
"We started turning over pucks," Jones said. "(And) then they started buzzing around."
Panarin made it 3-1 with his seventh of the season with 5:14 remaining in the third. New York's star left wing fired a shot from the left circle into an empty net as Lankinen had overcompensated for Strome at the goalmouth.
"That what he does," Gallant said about Panarin. "He always makes the right reads."
Panarin's goal capped a wild third period in which New York had two power plays and Chicago's Dominik Kubalik had a chance to tie the game at 2-2 but his semi-breakaway attempt hit the crossbar.
DeBrincat's 14th of the season with 1:20 remaining ended the scoring.
Prior to the game, New York announced Shesterkin had been placed on injured reserve with a lower body injury.
Shesterkin suffered the injury early in the third period of the Rangers' 1-0 win over the San Jose Sharks the previous night, and had to be helped off the ice by the trainer. He was replaced by Georgiev, who made nine saves to help complete the shared shutout.
During his pregame availability, Gallant said organizational decision-makers "feel comfortable" that the injury will not sideline Shesterkin for an indefinite period of time. He will miss New York's next three games, so Adam Huska was called up from AHL Hartford in the corresponding roster move.
Chicago interim coach Derek King announced defenseman Ian Mitchell had been called up from AHL Rockford due to Jake McCabe having "family matters" which required him to return to Chicago. King added "it was nothing really major."
Blackhawks: At the New York Islanders on Sunday night.
Rangers: At Chicago on Tuesday night to complete the season series with the Blackhawks.
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