After the event, Hegerberg told reporters that she accepted the apology from the 42-year-old musician.
Solveig was quick to apologise after his actions provoked furious criticism on social media.
Solveig was playing music throughout the ceremony, which included the men's Ballon d'Or and Kopa Trophy presentations, and also danced with Hegerberg to Frank Sinatra.
"I, of course, didn't want to offend anyone", he added."This comes from a distortion of my English culture level, which is obviously not enough because I didn't mean to offend anyone".
We should really be talking about Hegerberg's achievement.
The three-time Women's Champions League victor with French team Lyon says she'll be resting during the World Cup and training for the next season with her club.
Real Madrid's Croatia midfielder Luka Modric was named victor of the 2018 Ballon d'Or on Monday, breaking Cristiano Ronaldo's and Lionel Messi's decade-long hold on the prestigious award. One of her goals came in the final in Kiev in May, as Lyon came from behind to beat Wolfsburg 4-1 after extra time and retain the trophy.
Hegerberg called the award "a huge step for women's football" and also told The Associated Press she feels "very proud" to be the first Ballon d'Or-winning woman.
Twerk is used to refer dancing to popular music in a sexually provocative manner involving thrusting hip movements and a low, squatting stance.
That Solveig did so with Hegerberg immediately struck many as icky.
But Murray expressed his fury with Solveig over how the situation unfolded. "To everyone who thinks people are overreacting and it was just a joke, it wasn't". I've been invovled in sport my whole life and the level of sexism is unreal.
What should have been a historic moment in football was ruined when Ada Hegerberg was asked to twerk on stage at the sport's night of nights in Paris today.
In a second post, he said he had spoken to Hederberg and that she had understood the joke.
Hegerberg herself didn't let the uproar mar her evening. He came to me.
"I didn't take it as that at all", she said.
She cheerfully said, "I got to dance a bit and I got the Ballon d'Or". The woman's game has evolved over the years, but globally it should be valued more. She is already putting that leverage to use.
In an interview with the BBC later on Monday, Hegerberg sought to play down the moment.
Afterwards Hegerberg said it felt "incredible" to be the first female victor of the Ballon D'Or. "It's all about how we respect women's football".
We wish we could say we were surprised by the incident but, sadly, it follows a long history of women in football not being given the respect and recognition they deserve. A few hours later, Solveig's behavior made her words ring true. "Wow, men stay disappointing", said TBee.