Greater Manchester Police (GMP) said the 25-year-old is being held on suspicion of attempted murder after a couple were left with serious injuries, while a police officer was also stabbed as he tried to stop the suspect.
Prime Minister Theresa May joined British Transport Police in commending the emergency services for containing the threat during the incident, which occurred at around 9pm on Monday at Manchester Victoria Station.Assistant Chief Constable Russ Jackson, of Greater Manchester Police, said the suspect lived in the Cheetham Hill area of the city and counter-terrorism officers were at his address.
He said a couple aged in their 50s suffered multiple stab wounds in the "random" attack.
The suspect was heard to shout "Allah" and "Long live the Caliphate" during the incident at the station, which is next toManchester Arena where suicide bomber Salman Abedi killed 22 people on May 22 2017.
Mr Jackson said: "There is wide reporting in the press about what the attacker allegedly said during the incident and because of this we want to be clear, we are treating this as a terrorism investigation.
"However it's really important to stress we are retaining an open mind in relation to the motivation for this attack.
"We have nothing to suggest at this time others are involved.
"We believe we have identified the man in custody.
"In fact we are currently searching an address in the Cheetham Hill area of Manchester, we commenced that in the last 30 to 40 minutes.
"We know the attacker arrived at the location and soon after he attacked two people, a man and a woman, who have suffered very serious injuries.
"Whilst serious thankfully these are not life-threatening."
GMP Chief Constable Ian Hopkins added: "I know that the events of last night will have affected many people and caused concern.
"That the incident happened so close to the scene of the terrorist attack on May 22 2017 makes it even more dreadful."
A British Transport Police officer was also injured as officers on patrol "bravely and immediately" confronted the attacker with Tasers and pepper spray.
Mr Jackson said police are still trying to establish whether the suspect is a British national and how he came to be at the station.
He continued: "We are obviously considering his mental health given how frenzied the attack was, it's random nature but also consider the fact that it's at the same location as the Manchester (Arena) attack.
"We are exploring all avenues."
He said police recovered two knives but do not yet know if both were used in the attack.
Sam Clack, a BBC producer, was on a platform at Manchester Victoria waiting for a tram home when he witnessed the attack.
The 38-year-old 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-vis 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" officers jumped on the man and held him down.
He said he heard the suspect saying: "As long as you keep bombing other countries this sort of shit is going to keep happening."
Mr Clack said it looked like both the man in his 60s and the woman with him had been stabbed, but both were conscious and were walked to a waiting ambulance.
Police said there will be a greater police presence around the station on Tuesday, but added that there is no increased threat to people's safety.