The man, dressed in a black and red bandana, reportedly shouted about “Nazis of the UK media” as the singer performed her track Storm.
The intruder was dragged off stage by security after shouting “we demand freedom” and the singer clapped along until she was given a new microphone.
It came before the winner of this year’s song contest was announced as Netta Barzilai for Israel with her song Toy.
After SuRie finished singing, Graham Norton, who is the commentator for the BBC coverage, said: “She was doing a brilliant job anyway.”
He later added: “There was a stage invasion, SuRie coped incredibly well, I can’t stress enough what a brilliant job she did, she must have been terrified.”
The European Broadcasting Union offered SuRie the chance to perform again but she chose not to.
It said in a statement: “The EBU regrets a stage invasion took place during the UK performance at the Eurovision Song Contest Grand Final in Lisbon this evening. The person responsible is currently in police custody.
“The UK singer SuRie and her team were offered the option to sing again but decided not to because they were extremely proud of her performance and have decided that there is absolutely no reason to perform the song again.”
After the event, the 29-year-old, who came 24th out of 26 entries during the contest, tweeted: “Well, I’ve always said anything can happen at Eurovision …”
She said: “I’m so proud of my performance tonight. I gave it my very best.
“It’s been an incredible experience for me. I am so grateful for everyone’s love and support. The fans make Eurovision what it is; the biggest and the most inclusive and open celebration of music in the world.”
Celebrities and even MPs came out in support of the singer after her unfortunate experience.
JK Rowling wrote: “That was everyone who’s ever been on stage’s worst nightmare and it must have been scary, but what a pro, what a recovery @surieofficial.”
Former shadow home secretary Yvette Cooper added: “That was actually excellent #SuRie #eurovision – pumped it up even better for being interrupted.”
The man who interrupted the performance was taken into police custody after he was rushed off the stage.
A man dressed in the same bandana and t-shirt stormed the stage of the National Television Awards, where he was ordered off by host Dermot O’Leary.
Northern Irish talk show host Stephen Nolan also claimed the same stage invader crashed his programme last year.