Abe also called on UN member states to maximize pressure on North Korea until it implements UN Security Council decisions fully, agrees to change its policy, and sits on the dialogue table.
Hundreds of anti-North Korea protesters scuffled with riot police not far from the stadium before the opening ceremony began, with some banners reading "Moon regime is leading Korea to destruction".
The officials said it was fair to say both Pence and the North Koreans rebuffed South Korean President Moon Jae-in's efforts to facilitate a meeting between North Korea and the US.
Earlier on Friday, Kim Yo-jong, sister of the North Korean leader, arrived at Incheon International Airport along with a delegation headed by the country's ceremonial head of state Kim Yong-nam, 90.
"There is a faction that tries to turn this country communist.
If we don't stop it, we will be starved or tortured to death under the rule of Kim Jong-un", the organizers argued, in front of protesters wielding pickets that read "Moon Jae-in OUT" and "PyeongChang Olympics OUT".
Similar demonstrations took place sporadically around the city.
In a statement, the Organising Committee said the glitches "impacted some of our noncritical systems last night for a few hours" but insisted it had no affect on the Ceremony itself.
Kim Jong-un wanted to meet Moon "at an early date", a spokesman for the Blue House said. He told reporters that he would "continue to seize every opportunity to ensure that North Korea does not use the powerful imagery and backdrop of the Olympics to paper over an appalling record of human rights and a pattern of developing weapons".
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The visual was important, a USA official said.
South Korea showed off its technical expertise with a dazzling gala opening ceremony on Friday which included hi-tech special effects and augmented reality to add extra impact for TV viewers.
Despite hope that the Winter Olympic Games truce will last, many South Korean locals are more pessimistic, according to a United States report this week.
A lot of media organizations ignore this - mostly because the Associated Press stylebook tells them to, although CNN uses the capital C when referring to the PyeongChang 2018 Olympic Winter Games, not the town.
There were also lookalikes of US President Donald Trump and North Korea's leader Kim Jong-un gatecrashing proceedings.
Pyongyang has ruled out a meeting with the USA delegation during the Winter Games.
The North Koreans also said they were unlikely to have substantive discussions with Pence.
"The simple truth is that the time has come for North Korea to permanently abandon its nuclear and ballistic missile ambitions, to recognize there is no future as a member of the family of nations for a nuclear-empowered North Korea", he said.