The Dutch-American musician, 65, had been undergoing treatment for throat cancer.
His band, Van Halen, were best known for their song Jump, which hit the top of the US charts in 1984.
His son Wolfgang paid tribute to him on social media, saying he was the best father he could ever ask for.
“Every moment I’ve shared with him on and off stage was a gift,” Wolfgang, who became Van Halen’s bassist in 2006, wrote.
“My heart is broken and I don’t think I’ll ever fully recover from this loss. I love you so much, Pop.”
Some of rock’s biggest names have paid their respects to the “Mozart of rock guitar” on social media, with Kiss Singer Gene Simmons dubbing him a “guitar god”.
Celebrity news website TMZ, which first broke news of his death, said Van Halen died at St Johns Hospital in Santa Monica, California on Tuesday surrounded by his family.
The guitarist had been in and out of hospital in the past year and recently underwent a round of chemotherapy, TMZ reported, citing sources.
Van Halen is survived by his son Wolfgang and his wife Janie Liszewski, whom he married in 2009.
How did Van Halen rise to stardom?
Born in Amsterdam, Netherlands, Van Halen was the son of Eugenia Van Halen and Jan Van Halen, a clarinettist, saxophonist and pianist. The musical influence rubbed off on Van Halen, who was taught to play piano as a child.
When the family emigrated to Pasadena, California in 1962, they brought a piano on the boat.
“We actually played music on the boat on the way over here, you know? I’m serious! It wasn’t like, ‘so what do you want to do in life?’. Dad said, ‘we’ve got to make a living’. So if it weren’t for music, we wouldn’t have survived,” Van Halen said in a 2012 interview with Esquire.
As he grew up, Van Halen switched to drums and then guitar, initially playing alongside his father and brother at weddings and bar mitzvahs.
As an adult, Van Halen formed Van Halen in the early 1970s in Los Angeles. He was the guitarist, his brother Alex was the drummer, David Lee Roth was the singer and Michael Anthony was the bassist.
Van Halen was said to be the main song-writing force behind the band’s eponymous debut album, released in 1978. The album shot to number 19 on the Billboard charts, becoming one of the most successful debuts of the decade.
But Van Halen’s biggest success came with their sixth album, 1984, which saw Van Halen focus on synths over guitars. It produced the band’s only number one single, Jump, and the MTV favourite Hot For Teacher.
Jump was also the band’s only significant hit in the UK, reaching number seven in 1984.
That aside, the guitarist was possibly just as well known in the UK for playing the guitar solo on Michael Jackson’s 1983 hit Beat It.
Inducted into the Rock and Roll Hall of Fame in 2007, Van Halen received great acclaim for his distinctive guitar parts in many of the band’s hits, from Ain’t Talkin’ ‘Bout Love to Panama.
He refined and popularised a style of playing known as the “two-handed tap”; and his lightning-fast technique influenced countless rock guitarists in the 80s.
What do we know about his cancer battle?
Van Halen had been battling separate bouts of cancer for years.
He first confirmed he had tongue cancer in 2001, telling fans he would beat the disease. He received treatment and the subsequent surgery removed about a third of his tongue.
He was declared cancer-free in 2002. But TMZ reported last year that Van Halen had been privately battling throat cancer for five years, flying between the US and Germany for radiation treatment.
Throughout his treatment, Van Halen continued to attend concerts and rehearse music with his son Wolfgang, TMZ said.
Who has paid tribute to Van Halen?
Simmons tweeted: “My heart is broken. Eddie was not only a Guitar God, but a genuinely beautiful soul. Rest in peace, Eddie! …Eddie Van Halen Dead at 65 from Cancer.”
On Twitter, Red Hot Chili Peppers bassist Flea wrote that Van Halen had a “beautiful creative heart”.
Sammy Hagar, who replaced Roth as Van Halen frontman when he split from the band in 1984, shared a picture of himself alongside Van Halen, with the caption: “Heartbroken and speechless. My love to the family.”
Geezer Butler, founder member of Black Sabbath, described Van Halen as a “true gent and true genius”.
“Just when I thought 2020 couldn’t get any worse, I hear Eddie Van Halen has passed. So shocking – one of the nicest, down to Earth men I have ever met and toured with,” he wrote.
The Twitter account of rock group Pantera tweeted Van Halen was a “tremendous influence” on their music.
A post said: “RIP Eddie Van Halen. Van Halen were a tremendous influence on both Vinnie & Dime & Pantera. Hopefully they are all rocking out together now!”
Star Wars actor Mark Hamill sent his condolences to Wolfgang, adding: “But take solace in knowing that he will be long remembered and live on forever as one of Rock’s most extraordinary guitar virtuosos.”