In the run-up to the Games few expected the 26-year-old to challenge for a medal as all the focus was on Kenya's former Olympic champion Kiprop, who had won the last three world titles.
But after a pedestrian race that was won in a time 24 seconds outside world record pace, Centrowitz held off defending champion Taoufik Makhloufi of Algeria to cross the line in 3:50:00.
"It's been a dream come true," Centrowitz told reporters. "Doing my victory lap, I literally kept screaming to everyone I know 'Are you kidding me?'"
Britain's double 1,500m Olympic champion Sebastian Coe, the current IAAF President, awarded the gold medal to Centrowitz.
"Seb Coe came up to me and said 'Welcome to the club', which I thought was very cool," Centrowitz said.
Centrowitz, who won a world championship bronze in 2011 and silver in 2013, ran the final lap in 50.62 seconds to take gold in a field stacked with African talent.
New Zealand's Nick Willis, who took silver at Beijing in 2008, overcame several injury-plagued seasons to snatch a bronze medal in his fourth Olympic games.
"After 2008, I had three different surgeries, it took me four years to get back into form," the 33-year-old Willis said, hinting that he may compete at the next Olympics in Tokyo.
"A lot of people write you off in your 30s. This is for all the 30-year olds."
Kiprop, who had run the fastest time of the year, faded in the final lap to finish sixth. His compatriot Ronald Kwemoi took a tumble early in the race and Kiprot said that affected his tactics.
"When I saw Kwemoi floored on the track, I kind of felt very lonely," Kiprop said.
But Willis, who has often raced Kiprop, was not surprised by the result.
"Having been around with Kiprop since 2007 ... what's most consistent about Kiprop is his inconsistency," Willis said.
"He is extremely talented and probably the best in the world, but can also lose concentration."
(Additional reporting by Isaack Omulo; Editing by Ed Osmond/Greg Stutchbury)
