British wild card Broady shocks world number four Rudd
Broady held his nerve when trailing two sets to one against the French Open finalist, rallying to win the Centre Court clash 6-4, 3-6, 4-6, 6-3, 6-0.
Norway's Ruud, who has been runner-up at Roland Garros in the past two years, was also the beaten finalist at the US Open last year.
The 24-year-old has never gone beyond the second round at the All England Club.
Broady, 142 in the world rankings, took a medical time-out at the end of the fourth set to change some bandaging on his wrist.
But the 29-year-old came out fit and firing in the evening sunshine, demolishing Ruud 6-0 in the decider before soaking up the applause from the partisan crowd.
"I would have liked to have played him back home, but Centre Court of Wimbledon will do," said Broady.
"When I went to bed last night, I was having a think of what I'd say if I won the match, but now I'm here I don't know what to say.
"I said to my mum this morning, she doesn't like watching, but I said I've already won £80,000 this week so she can chill out a bit."
The British player will play Canadian 26th seed Denis Shapovalov in the third round.
"Denis is a mercurial talent, he's one of the best players in the world as is Casper," he said. "He's won junior Wimbledon. He's an amazing player. With a crowd like this then why not have a go again."