Medvedev, who also beat Musetti at last week's Canadian Open before losing in the quarter-finals, broke the Italian twice in each set and won 82% of his first-serve points en route to sealing victory in 77 minutes.
"We both made some mistakes, we both made some double faults and stuff like this," third seed Medvedev, who served five double faults, said during his on-court interview.
"But I managed to be a little bit better in the important moments and I felt like I was playing better and better during the match so that's a positive thing and I'm looking forward for the next round."
Medvedev wobbled early as he failed to hold serve in the opening game but it did not unsettle the tall Russian as he responded immediately with a break at love and struck again in the eighth game before serving out the first set.
Musetti did well to claw back from 0-40 down to hold for a 2-1 lead in the second but Medvedev responded in style with a love hold and broke the Italian's serve on the next two opportunities before closing out the match.
In other second-round action, Italy's Jannik Sinner (22), who won the Canadian Open, faces Serbian qualifier Dusan Lajovic (33) and fourth seed Stefanos Tsitsipas (25) meets American Ben Shelton (20).
Djokovic, hoping to draw level with Margaret Court's all-time record haul of 24 Grand Slam titles by winning the US Open, kicks off the night session against Spain's Alejandro Davidovich Fokina (24).
