The easy victory got the 1997 champion back into her stride on the All England Club grass after her upset 12 months ago when she was defeated in the first round by Australian qualifier Jelena Dokic.
Montolio, playing only the second Wimbledon match of her career, looked completely outclassed until the final game when she produced her best tennis of the match, saving seven matchpoints.
Top seed Hingis first served for a place in the second round with 45 minutes on the clock.
But Montolio, 5-2 down after dropping her serve twice, prolonged the action for another nine minutes as she battled to stay in the match.
The court one game went seven times to deuce as Hingis began to struggle with her first serve and with her concentration, which was clearly upset by a foot fault called on matchpoint number five.
Montolio, the world number 42, hit a service return out to give Hingis her eighth matchpoint and then put a backhand long to concede victory.
``I just couldn't finish if off,'' said Hingis, 19. ``She played so well on those matchpoints. Maybe I didn't serve so well and she always came up with an amazing winner.''
The Swiss 19-year-old said last year's defeat, which came after she had quarrelled with her mother who has been the guiding light of her career, had humbled her.
``I think I learned a nice lesson here which was positive. Sometimes you have to experience things. People go to school and do worse things than I did last year.''
Hingis patched things up with her mother, Melanie Molitor, after Wimbledon and this year has reached the final of the Australian Open and the semifinals of the French.
``A few weeks after Wimbledon I won the tournament at San Diego and I kept on going in the right direction,'' she said.
Hingis should find it easy to keep on in that direction. She now faces either China's Yi Jing-Qian, the world number 78, or Italian qualifier Giulia Casoni.