German police in the city of Konstanz said they were awaiting the extradition of the 19-year-old from the nearby Swiss city of Kreuzlingen.
The Cologne attacks, committed in a crowd of mostly North African men, shocked Germany and heightened public concern about an influx of migrants, many of them Syrians fleeing their country's civil war.
Prosecutors received more than 1,100 criminal complaints, including over 480 accusations of sexual assault.
The attacks sparked widespread anger, with far-right groups railing against "sex jihadists" and "rapefugees".
The 19-year-old suspect was arrested on Wednesday after he and a 24-year-old man were allegedly spotted stealing cigarettes and groceries worth €24 from a supermarket in the frontier city of Konstanz.
An undercover security worker at the shop followed them on foot across the Swiss border, then contacted German police, who in turn alerted the Swiss Border Guard.
"One of the border guards present thought he recognised the [19-year-old] man as an alleged main perpetrator of the events of New Year's Eve in Cologne," said a German police statement.
"Subsequent verification of his personal details by German authorities revealed that the 19-year-old was indeed subject to a European arrest warrant due to these events issued by Cologne prosecutors."
Kreuzling prosecutors had since issued an extradition warrant, said the statement.
A 33-year-old Moroccan went on trial earlier this month in Düsseldorf accused of sexual assault during New Year's celebrations in that city.
In Cologne, three people were convicted of theft during the year-end celebrations, but police there have admitted that most perpetrators of the sexual attacks may never be caught.