She was attacked after being forced into an alleyway off Borough Road, near to Tranmere Rovers' stadium in Wirral, at about 05:30 BST on 28 September.
The man ran off and she sought help at a nearby house. Police were contacted after she went to hospital for treatment.
CCTV images have been released of a man who police want to speak to as part of their investigation.
The woman had shared a taxi to Wirral from Liverpool city centre with a friend and a man she did not know, police said.
Her friend got out at Whetstone Lane, Birkenhead, before the woman and the man got out in Borough Road to use a cash machine.
She was then forced into an alleyway where she was attacked, police said.
An 18-year-old woman has been raped following a night out in Merseyside.