Police declared a major incident in Tameside at about 23:45 GMT on Wednesday "due to the severity of the damage caused and potential risk to public safety".
About 100 homes were damaged in the storm and people have been asked to avoid the area.
There have been no reported injuries.
Elsewhere in the UK Storm Gerrit has brought flooding and disrupted travel, with Scotland worst affected.
The north-west of England, the southern coast, Wales and Northern Ireland were also covered by Met Office yellow weather warnings on Wednesday, with heavy rain and wind battering them.
The Liberal Democrats have called on Rishi Sunak to convene a Cobra meeting in the wake of Storm Gerrit and the tornado in Greater Manchester.
The Met Office said the storm had now passed but further unsettled weather was likely later in the week, with strong winds, rain and some upland snow
In Tameside, the Millbrook and Carrbrook areas in Stalybridge were worst affected by the heavy rain and wind.
Millbrook resident Maisie Pilkington said a tree came through her roof as she slept.
"I opened the door and thought 'oh my God'," she said.
"The tree's hit the roof and gone through my bathroom.
"The whole ceiling of my bathroom is down, the tree's in the bathroom and the whole tree is being held up by one branch.
"If it snaps, it's going to go through the whole house."
Stalybridge resident Kerry Hogan told BBC Radio 5 Live the tornado "lasted a minute, 90 seconds, literally a whirlwind of madness".
"We couldn't open the front door as the wind was that bad," she said. "It sounded horrendous, it was very, very loud.
"Lamp-posts were swaying, we watched trees topple like dominoes.
"All the neighbours were out, it was bedlam.
"People have had cars crushed.
"It was horrific and how nobody has been hurt is unbelievable."
Carrbrook resident John Spence told BBC Radio Manchester: "I thought the windows were going to come in."
He said he had never experienced anything like it.
"I was scared of going out," he said. "It snapped my flagpole like a twig and took a tree down in my back garden that fell on the conservatory."
Millbrook resident Leah Solomon said the wind was so strong, she thought her windows "were going to come through".
She said the worst of the storm lasted for about a minute "and then all the alarms started".
"All the neighbours came out and there was just so much damage," she said.
"It was really frightening and like nothing that I've ever seen before."
Other parts of the UK have been hit by Storm Gerrit, with flooding and disrupted travel, including:
BBC weather presenter Matt Taylor said there had been a "big clutch of thunderstorms" across Greater Manchester and the Met Office said that "it is likely that it was a tornado which brought the damage to Stalybridge".
"It was just a small, short-lived tornado and that's why the damage was limited to quite a small area," he said.
Fire and ambulance crews joined officials from Electricity North West and the council at multiple sites across Tameside. A rest centre was set up at Dukinfield Town Hall.
A tree crashed through a roof on Grenville Street in Millbrook, with residents evacuated by the emergency services late at night.
In the daylight they were confronted with the impact of what had happened and were waiting by the police cordon to find out when they would be allowed back inside.
Residents said their houses shook and they heard a loud roaring sound and they feel they were lucky they were not hurt.
They think it only lasted for about 30 seconds, but in that short time, the lead lying on the ground, which is too heavy to lift, was blown off their roofs and onto the cars below.
Mr Taylor added: "What is quite surprising to many is that we actually see more tornadoes per square mile here in the UK than anywhere else in the world.
"But like we saw in Stalybridge they're usually very small, they don't last long, and so we don't really notice them as much as you do in other parts of the world.
"Tornadoes, whatever their size, can cause huge amounts of damage. You can see short and longer spells of very intense winds of over 100 miles an hour. Hence you can see damage to buildings, to property and trees and various other structures brought down quite quickly."
Greater Manchester Police said the damage had been caused "due to a localised tornado".
Ch Supt Mark Dexter said: "This incident has undoubtedly affected numerous people in the Stalybridge area with many residents displaced from their properties during the night.
"Our highest priority is keeping people safe which is why we are advising those who have been displaced not to return or enter their properties which have significant damage until they have been assessed by structural engineers."
He added: "I would also like to urge members of the public to avoid the area where possible and take extra care when travelling in vehicles on the roads in Stalybridge and the surrounding areas, due to debris in the road.
"This has understandably caused some disruption and, though we are not yet in a position to confirm when the area will return to normal, further updates will be communicated when we have them."
Greater Manchester Fire and Rescue Service said crews were called "to multiple reports of damage to various sites and properties in Stalybridge at about 23:00 and worked to make homes and residents safe until 07:00".
By Emma Stanley & Fiona Trott in Stalybridge, BBC News, 29 December 2023