Evenepoel was forced to switch bikes on a climb 50km from home but showed all the power that brought him the world and Olympic time-trial titles to rejoin the leaders with 33km remaining.
The Belgian then attacked his fellow leaders on the final short Murgil climb, with only Carapaz able to follow.
Evenepoel had enough left in the tank to win by a bike length in downtown San Sebastian as he made history in the 221km Basque one-day classic.
"It was the most difficult one of all four," said the 26-year-old, describing it as a beautiful day.
"I had to be patient. I have to thank my team for all the work, of course," he said of his Red Bull-Bora Hansgrohe teammates and their efforts to shepherd him to the final hill.
Carapaz won the King of the Mountains polka-dot jersey at the recent Tour de France while Evenepoel finished second overall to Tadej Pogacar.
Time-trial specialist Evenepoel is also the reigning Olympic road race champion and said he would now turn his attention to trying to reclaim the world road race title he won in 2022.
"First, what is on my mind is some recovery, some rest, and then I'm going to build up to try and become world champion," he said.
Reigning two-time world champion Pogacar will also be in Canada to defend his title on September 27.