The 22-year-old man is currently detained in police custody in connection with the incident. A report will be submitted to the Procurator Fiscal and he is expected to appear at Glasgow Sheriff Court on Wednesday.
A 24-year-old woman had been walking along Ingram Street before the incident happened in a basement area.
Earlier, a charity warned that everyone must remain vigilant to ensure the prevention of rape - not just women - following a string of other sex attacks in the city.
Hundreds of people took to the streets in Glasgow's south side at midnight last night in a protest against the attacks.
Police Scotland said it was in attendance at the event and officers were available to speak to anyone with concerns.
Rape Crisis Scotland said urging women to keep themselves safe did not go far enough to address the problem.
Police are stepping up patrols in the city after two other women were allegedly raped and two sexually assaulted in a spate of unrelated attacks.
Officers warned people to remain vigilant and be aware of sexual predators who prey on the vulnerable,
However, a spokeswoman for Rape Crisis Scotland said more had to be done to create a shift in attitudes to sexual violence.
She said: "Focusing on women's safety in the aftermath of a series of sexual attacks such as we have seen in Glasgow is a natural response.
"However it's important that as we do so we do not simply talk about 'women keeping themselves safe' and place the onus on them to restrict and limit their behaviour.
"It's vital to acknowledge that this is everyone's responsibility and discuss things like prevention initiatives which address and challenge the attitudes which lead to sexual violence, and bystander training to enable people to own and act on their share of that responsibility."
The charity said there were a number of initiatives running to train people in how to deal with a situation where they suspect someone is in a vulnerable position, for example a man who appears to be preying on a woman who is very drunk.
One such initiative is run by Police Scotland's Violence Reduction Unit and involves officers training bar staff, door stewards and other workers to be on the lookout for the vulnerable.
The spokeswoman added: "This training can be very effective. It basically allows people to intervene in a situation where they see somebody in a vulnerable position.
"It's also important that we raise awareness around consent and attempt to address these misogynistic attitudes and look at where they're coming from.
"We need to look at the whole picture. While there's always a place for public safety initiatives, it's not just about women looking after themselves."
Officers are hunting two men who raped a 24-year-old woman in Govanhill a few weeks ago on Monday, May 19.
It followed the gang rape of another woman in Toryglen in the same month.
Sex attacks have also taken place at Glasgow Green on April 19 and Kelvingrove Park on May 30.
Detective Superintendent Peter McPike said: "I am aware that the recent incidents are causing concern. However, I would like to reassure people that officers investigating these crimes are specially trained to deal with these particular circumstances.
"Rape is a devastating crime and has a significant impact on victims and their families. It also impacts on our local communities. Over the past few months, we have had, and will continue to have, additional patrols in these areas providing reassurance to local people. Anyone with any concerns should raise them with these officers.
"It's not acceptable that people, especially women, should live their lives in a state of perpetual vigilance or be responsible for the actions of others. Our message is clear that responsibility for sexual violence lies with the perpetrators and this predatory and exploitative behaviour has to stop."