King kicked the ball from his own 6-yard box, and the wind escorted it down the pitch and into Joshua Griffiths' goal.
Begovic's 100.5-yard (91.9-metre) long-range effort had held the record for seven years. He scored against Southampton in the Premier League on Nov. 2, 2013.
"I'm absolutely delighted, because it's not something that I intentionally set out to do," King said in a statement.
"I'm sure it will be talked about for a long time to come, so I'm proud, and I'm sure my family will be extremely proud too.
"I'll have to drop Asmir a message to send my commiserations, but, in all honesty, it's different territory for me because I never dreamt of being in the books that you get for Christmas every year.
"Hopefully I might actually read it if I get one for Christmas this year. I'm delighted, and hopefully no one beats it for a long time now so I can show my grandkids!"
Newport County are second in League Two with two games in hand over first-place Cambridge United.
King is the club's first-choice goalkeeper after joining the side in 2019. The same year, he was called up to the Wales national side.
Newport County pulled off a shock 3-1 over Championship side Watford in the Carabao Cup in September.