Four-year-old Cleo Smith, who went missing from a campsite in Western Australia more than two weeks ago, has been found alive and well, police said on Wednesday.

WA police said they had found Cleo in a house in Carnarvon about 1am on Wednesday and a local man had been taken into custody.

A deputy commissioner, Col Blanch, said police had broken their way into the locked house and found the girl in one of the rooms.

“It’s my privilege to announce that in the early hours of this morning, the Western Australia police force rescued Cleo Smith,” he said. “Cleo is alive and well.

“One of the officers picked her up into his arms and asked her, ‘What’s your name?’

“She said, ‘My name is Cleo.'”

Cleo was reunited with her parents a short time later.

The four-year-old vanished from the Blowholes campsite near Carnarvon, about 950km north of Perth, on 16 October.

Her mother, Ellie Smith, was the last known person to see her. She said her daughter woke about 1.30am asking for water before going back to bed. In the morning Cleo and her sleeping bag were gone from the family’s tent.

“This is the outcome we all hoped and prayed for,” Blanch said.

The Carnarvon shire president, Eddie Smith, told 2GB radio on Wednesday that he knew something was going on when his phone started ringing.

“My phone started to ring flat out and I knew there was something going on here. Cleo Smith had been found. I rang the police and they told me it was correct.”

Australia’s leaders were quick to express their joy at the news and to thank the police and others who had searched for Cleo over two harrowing weeks.

The prime minister, Scott Morrison, said on Twitter it was “wonderful, relieving news”.

“Our prayers answered. Thank you to the many police officers involved in finding Cleo and supporting her family.”

The Labor leader, Anthony Albanese, said it was “such happy news”.

“To all those who tirelessly searched for this little girl, you deserve our thanks.”

Blanch said the outcome was down to “some incredible police work”.

Sign up to receive an email with the top stories from Guardian Australia every morning

“I want to thank Cleo’s parents, the Western Australian community and the many volunteers. “And of course, I want to thank my colleagues in the Western Australia police force.”

The NSW police commissioner, Mick Fuller, told 2GB he broke down and cried when he received a call with the news from his WA counterpart, Chris Dawson.

Fuller said had thought the chances of finding Cleo alive were “so slim”.

“West Australian Police never gave up on Cleo, it was good old fashion police work that resulted in her being found alive,” Fuller said.

Blanch said the man in custody was being questioned by detectives.

“We’ll have more to say on the rescue of Cleo as the day unfolds,” Blanch said. “For now – welcome home Cleo.”

A press conference is expected to be held mid-morning Perth time.

Detective inspector John Rouse, the manager of victim identification at the Australian Centre to Counter Child Exploitation, said he emailed colleagues in WA early on Wednesday morning to congratulate them for recovering Cleo. Rouse said he had seen the heartbreak of when children cannot be recovered, including the tragic case of Queensland boy Daniel Morcombe, who was abducted from a bus stop in 2013.

A taskforce of 100 police officers, led by Det Supt Rod Wilde, has been based out of Carnarvon for 18 days looking for clues for her disappearance. They used reconnaissance planes to carry out real time surveillance.

The position of the zipper on the tent flap – too high for Cleo to reach – suggested she had not opened the tent herself.

“The positioning of that zipper for the flap is one of the circumstances that has caused us to have grave concerns for Cleo’s safety,” Inspector John Mundy said five days into the search.

That prompted police to pursue the “nightmare scenario” that she had been abducted.

Cleo’s mother maintained throughout the ordeal that Cleo would never leave the tent on her own.