Wales started their Euro 2016 campaign with a win over Slovakia but Group B rivals England were held by Russia.
Violence marred the run-up to England’s 1-1 draw in the French city of Marseille and flared again afterwards.
Wales marked their first appearance in a major tournament since 1958 with a 2-1 victory in Bordeaux, Gareth Bale and substitute Hal Robson-Kanu the scorers.
Northern Ireland play their first Group C game on Sunday against Poland in Nice (17:00 BST).
In Marseille, Tottenham’s Eric Dier put England ahead in the second half but they were denied victory by a stoppage-time equaliser from Vasili Berezutski.
England boss Roy Hodgson told BBC Radio 5 live: “When you concede with one minute left, it is a bitter disappointment.
“In the first half I thought that was as good a performance as we are capable of giving.
“It feels like defeat because we were preparing to celebrate a victory and we don’t have that possibility now.”
The end of the game was overshadowed by more clashes between rival fans.
Russia supporters appeared to rush at the England contingent, and BBC Radio 5 live’s Mark Chapman said: “The mood in the stadium had been fine until about five minutes to go – at which point two flares were set alight in the Russian end.
“It then looked like a firework was launched from that end into the night sky and the mood in the stadium then notably changed.”
The latest violence followed two nights of trouble in the French port city, where tear gas and a water cannon have been used by police.
Earlier on Saturday, around 24,000 Wales fans were at the Stade de Bordeaux as Chris Coleman’s side claimed a famous win.
Playing in their first major tournaments for 58 years, Wales took the lead through Real Madrid’s Bale before Robson-Kanu’s scuffed finish won it for Wales after Ondrej Duda had equalised for Slovakia.
“Our fans are the best in the world. It was like a home game – they fully got behind us and we gave them something to celebrate,” said Bale.
“It was a historic moment for our country, a massive moment.”
England and Wales play against one another in Lens on 16 June (14:00 BST).
McIlroy gets behind Northern Ireland
The other home nation in France, Northern Ireland, are tournament outsiders but have the longest unbeaten run of any of the 24 competing nations.
Michael O’Neill’s men have gone 12 games without losing, their last defeat coming in a friendly against Scotland in March 2015.
“I have never prepared a team as well as this,” said 46-year-old O’Neill. “It’s exciting to be here after four weeks of preparation.”
Northern Ireland golfer Rory McIlroy, in a video message, urged the national football team to “write the next chapter” in the country’s sporting story.