Raonic, who has fallen from No. 3 to No. 30 due to numerous injuries the past two seasons -- including a quadriceps tear at Wimbledon last month -- used his powerful serve to his advantage, firing 13 aces to Goffin's two, and won 100 percent of his first serves.
"I think I can still serve much better, I don't think I served particularly well," Raonic said. "So I'll take the time to work on some things tomorrow but overall it was a good performance. Mentally I was in the right state of mind the whole way through and I was very disciplined with myself."
Shapovalov and Auger-Aliassime started the match strong before Djokovic and Anderson took control. They broke Anderson's serve for a 2-0 lead, to the delight of a tightly packed grandstand crowd. However, the Wimbledon finalists team -- nicknamed Djokerson thanks to a Twitter poll conducted by Djokovic earlier in the day -- broke back to tie the match 2-2, and again to go up 5-3.
Anderson and Djokovic won five consecutive games, going up two breaks, to win the second set.
"Our game was there, we didn't feel intimidated at all," the 19-year-old Shapovalov said.
"Just to have a chance to play with these guys is already good," added Auger-Aliassime, who won't turn 18 until later this week.
In other singles matches, American Bradley Klahn topped Spain's David Ferrer 7-6, 6-4; Pierre-Hugues Herbert of France got past Spaniard Albert Ramos-Vinolas 4-6, 6-3, 6-4; Benoit Paire of France defeated Jared Donaldson 6-3, 6-4; Ilya Ivashka of Belarus beat Yuichi Sugita of Japan 6-2, 6-3; and Borna Coric of Croatia was a 6-4, 6-3 winnner over Canadian Vasek Pospisil in the late match on center court.
Also, 30-year-old Canadian Peter Polansky, a wild-card entry playing in the morning draw, defeated Matthew Ebden of Australia 7-6 (3), 6-4. He will play the winner of a match between Djokovic and Hyeon Chung of South Korea in the second round.