In a nail-biting end that had the Le Bourget crowd on the edges of their seats, Anraku, the last to go in the competition, slipped just short of the hold on the lead wall that he needed for the title.
His premature fall confirmed Roberts's win, crowning him the Olympics' first boulder and lead champion and prompting him to raise his hands to his head in sheer disbelief.
"To be honest I literally have no words. I don't think this is ever going to sink in," he said.
Asked about the moment he realised he had come out ahead, he said: "I was happy with the silver so I was just sat there completely happy and then to find out that I got gold, it was just an incredible moment."
With a total of 155.2 points from the combined boulder and lead events, Roberts overtook Anraku's 145.4. Austria's veteran climber Jakob Schubert earned a back-to-back bronze with 139.6.
Sport climbing is making its second Olympic appearance but the Paris Games are the first where the combined boulder and lead event has been split from speed as is the norm for the sport.
In boulder, athletes have a time limit to negotiate four boulder 'problems' on a 4.5-metre-high wall, earning points on the highest-scoring holds they manage. In lead, they aim to climb a 15-metre wall as high as possible in a single attempt.
