Raonic faces 2007 winner and world No. 4 Andy Murray in the third round after the Scot opened his tournament with a 6-2, 6-3 win over Alejandro Falla.
Raonic, ranked 26th in the world and seeded 26th at the tournament, finally got the win with a break after Clement had saved three match points on his own serve.
The hard-serving 21-year-old from Thornhill, Ont., finished one hour 43 minutes match with 17 aces and three breaks of 153rd-ranked Clement.
"I played well in the first set, and I dropped off a little bit at the end of the first set," said Raonic. "Just too many mistakes that one game I got broken. In the second set I just cleaned up a few things and they just came together for me."
Meanwhile, on the women's side, Canada's Aleksandra Wozniak rallied to upset Monica Niculescu 4-6, 6-3, 6-4 to also move on to the third round.
The wild-card entry from Blainville, Que., fought back after trailing 2-4 in the final set to down the world No. 28.
Top-ranked Victoria Azarenka extended her winning streak to 24 matches by beating Michaella Krajicek 6-3, 7-5.
Azarenka's streak began with her first match this year, and her start is the best since Martina Hingis went 37-0 to begin 1997. Azarenka is bidding for her third Key Biscayne title and second in a row.
"I'm not really thinking about numbers, the streaks, whatever," Azarenka said. "It's your job to count; mine is to play tennis."
No. 15 Ana Ivanovic lost only six points on her first serve and beat Vania King 6-4, 7-5. Ivanovic was playing for the first time since she retired with a left hip injury a week ago at the Indian Wells semifinals.
Another wild card player stunned a ranked opponent as Garbine Muguruza Blanco upset No. 9 Vera Zvonareva 6-4, 6-3.
In other action, No. 18 Angelique Kerber lost to Zheng Jie 7-6 (4), 6-1 and No. 31 Kaia Kanepi was beaten by Silvia Soler-Espinosa 7-5, 6-3.
Also, Venus Williams took a big step in her comeback when she defeated reigning Wimbledon champion Petra Kvitova 6-4, 4-6, 6-0 in the second round.
Williams is playing in her first tournament since she withdrew from the U.S. Open in August after being diagnosed with a fatigue-causing autoimmune disease.
She showed plenty of staying power against the third-seeded Kvitova, taking charge in the final set. When Kvitova pushed a weary forehand wide on the final point after nearly 2 1/2 hours, Williams trotted to the net with a grin and raised fist.
A former No. 1 and seven-time Grand Slam singles champion, Williams is ranked No. 134 and needed a wild card to enter the tournament, which she has won three times.
Murray, who has never faced Raonic, bounced back from a loss in his first match at Indian Wells two weeks ago. He also avoided a repeat of last year, when he lost his opening match at both Indian Wells and Key Biscayne. With the victory, the No. 4-seeded Murray improved to 15-3 this year.
"Obviously I have to serve well, take care of my serve," said Raonic. "That's my job against everybody I play.
"I'm fortunate enough to have the game style to impose and to dictate. So I don't feel like I need to change too much. I just need to clean up a few things, hopefully do them well, and I know if I do that I will have my opportunities."
Raonic was defeated in the second round at Indian Wells by eventual champion Roger Federer.
Other seeded men to reach the third round Friday included No. 2 Rafael Nadal, No. 6 Jo-Wilfried Tsonga, No. 9 Janko Tipsarevic, No. 10 John Isner, No. 13 Gilles Simon, No. 18 Alexandr Dolgopolov and No. 22 Jurgen Melzer.
Wozniak, ranked 65th, has never been past the second round at the tournament. She levelled the third set 4-4, then broke Niculescu before serving out the victory on the first of two match points.
Wozniak won a lower-level title last week in the Bahamas after exiting the Indian Wells tournament in the first round earlier this month.
"It was a tough match overall, she played well," Wozniak said. "I had to adjust and fight in the third set. I kept pressing in the third and missing less.
"I've won seven matches in a row now. You really need matches to build your momentum and your game."
Azarenka double-faulted six times and lost three service games, but Krajicek was even more erratic with her serve. Krajicek failed to hold at 5-4 in the second set, double-faulting three times, and Azarenka swept the final 10 points.
The Belarusian seeks her fifth title this year after winning at Indian Wells a week ago.
"It was important to raise my level at the end of the match," Azarenka said. "I found the right shot selection and the right opportunities when I needed to."
With files from The Associated Press.