The suspect is being held by Greater Manchester Police on suspicion of attempted murder following the attack at Victoria Station at around 9pm on Monday.
Two commuters - a man and woman in their 50s - were taken to hospital with knife injuries and a British Transport Police (BTP) officer was stabbed in the shoulder. He has since been released from hospital.
A GMP spokesman said: "The police sergeant was taken to hospital with a stab wound to his shoulder following the attack at the station.
"Within minutes, a man was arrested near to Metrolink platforms A and B on suspicion of attempted murder. He remains in custody for police questioning."
The station is next to Manchester Arena, where suicide bomber Salman Abedi killed 22 people on May 22.
The spokesman said there would be a greater police presence around the station on Tuesday, but added that there was no increased threat to people's safety.
A witness to the attack said they heard the assailant shout "Allah" during the incident.
Assistant Chief Constable Rob Potts said: "(Monday night's) events will have understandably worried people but I need to stress that the incident is not ongoing, a man is in custody and there is currently no intelligence to suggest that there is any wider threat at this time."
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.
He said he also heard the knifeman shouting "Allah" during the attack.
"He shouted it before, he shouted it during it - Allah," Mr Clack said.
As police appeared "within seconds", the knifeman backed down the platform towards where he was standing alone.
Mr Clack described how he feared the man would attack him.
"I just had a feeling in the pit of my stomach," he said.
"Here's a man who's very agitated, angry, skittish. It appears he had already attacked someone. He had police coming towards him and he's coming towards me.
"I thought what I need to do is jump on to the tracks because next thing he's going to do is turn around and stab me. He's looking about, side to side and jumpy.
"At one point one of the police officers fired the Taser but it missed him.
"Then he got pepper sprayed and Tasered and he want to the ground.
"The guy, his exact words were, he said: 'As long as you keep bombing other countries, this sort of shit is going to keep happening'."
He added: "It was scary. I have never been so scared in my life. Someone with a knife six to eight feet away, he had just stabbed someone. It was the proximity.
"It just highlights the fact that it can happen anywhere.
"It's just a guy with a knife on a platform. It can happen anywhere. Anyone can do it."