A 25-year-old man was arrested on suspicion of attempted murder after the attack, which happened shortly before 9pm.
Two knives were recovered at the scene and a property is being searched in the Cheetham Hill area.
In a briefing on Tuesday morning, police said they were keeping an open mind about the motive.
They have not yet confirmed whether the suspect is a British national.
The three victims were taken to a nearby hospital in a "serious" condition but their injuries are not thought to be life-threatening.
A woman in her 50s was stabbed in her head and stomach, while a man - also in his 50s - was stabbed in his stomach.
They are still being treated in hospital.
A British Transport Police sergeant in his 30s was also treated after he was stabbed in the shoulder, but has since been released.
British Prime Minister Theresa May tweeted today: "My thoughts are with those who were injured in the suspected terrorist attack in Manchester last night. I thank the emergency services for their courageous response."
Sam Clack, a BBC producer who works in the city, was on a platform at Manchester Victoria when he witnessed the attack.
He said: "I just heard this most blood curdling scream and looked down the platform.
"What it looked like was a guy in his 60s with a woman of similar age and another guy all dressed in black.
"It looked like they were having a fight, but she was screaming in this blood curdling way. I saw police in high-viz come towards him. He came towards me.
"I looked down and saw he had a kitchen knife with a black handle with a good, 12-inch blade. It was just fear, pure fear."
Mr Clack said police officers used a Taser and pepper spray before "six or seven" of them jumped on the man.
A New Year's fireworks display in the city's Albert Square went ahead with "increased security".
Ian Hopkins, the chief constable of Great Manchester Police, called the incident a "horrific attack on people simply out to enjoy the New Year's Eve celebrations".
He said British Transport Police officers had showed "incredible bravery" in tackling and detaining the suspect - who remains in custody in Manchester."
