The company said it "fell well short" of its goal to ensure cards work reliably at all times following problems on Friday afternoon.
Several bank machines were out of order on West Street in Sheffield yesterday afternoon though it is not currently known if this was related to Visa's system failure.
Visa users complained about the issue, which was the result of a hardware failure, on social media.
A Visa spokesman said on Friday night: "Earlier today, Visa had a system failure that impacted customers across Europe.
"Our goal is to ensure all Visa cards work reliably 24 hours a day, 365 days a year.
"We fell well short of this goal today and we apologise to all of our partners and, most especially, to Visa cardholders."
The payment system was said to be operating at "close to normal levels" by 10.30pm on Friday, but Visa Europe posted a statement on its website in the early hours of Saturday to say the problems had been resolved.
It said: "Visa Europe's payment system is now operating at full capacity, and Visa account holders can now use Visa for any of their purchases and at ATMs, as they normally would."
The issue was not associated with any unauthorised access or cyber attack, the statement added.
It is understood the Bank of England was in touch with Visa about the problem.
Alex Neill, Which? managing director of home products and services, said the issue would have been a "huge inconvenience to customers" and said Visa and the banks now need to ensure no-one is left out of pocket.
She added: "We strongly advise people to keep any evidence of extra expenses they've incurred in order to claim them back."