Officers were called to the home on West First Street in Bayonne at 6:45 a.m., and a short time later, shots rang out inside.
The initial stabbing was reported to be on the third floor, but so far, there has been no update on the condition of the person stabbed or the extent of any injuries that person may have suffered.
Authorities say the officers encountered the man, when one or more officers fired their service weapons.
MORE NEWS | NJ high school student dead, another hospitalized after shooting at field
Officials with Bayonne Medical Center said a man in his 40s arrived with multiple gunshot wounds, and despite the efforts of the emergency room staff, he succumbed to his injuries.
"The medical staff provided him with the best medical care and trauma protocols," the hospital said. "Despite their efforts, he did not make it."
It appeared the person with the knife had been dealing with mental health issues, and neighbors said he was known in the area.
"I know he had some mental issues -- that's what we did here a while ago, that he had mental issues, but you never know," Bayonne resident Colleen Higgins said.
They also said the woman living in the home with her son was a retired crossing guard.
"She's a sweet old lady, and I didn't know her boys -- but the one boy died last year and now this one, it's a shame," Bayonne resident Madeline Wallace said.
The incident happened on a quiet residential block overlooking the waterway between Bayonne and Staten Island.
The block has now been closed off to all traffic, as police and investigators remain on the scene.
As far the officer involved in the shooting, he has been treated and released.
The case has been turned over to the state attorney general for a full investigation.
MORE NEWS | Actress Lisa Banes in critical condition after hit by scooter at Manhattan intersection
* Get Eyewitness News Delivered
* More New Jersey news
* Send us a news tip
* Download the abc7NY app for breaking news alerts
* Follow us on YouTube
Submit a News Tip