Fresh from winning the French Open just a few weeks back, the 31-year-old romped to a 6-1 6-3 victory against her opponent from Luxembourg in just 57 minutes.
Williams may have wished she could have spent a little longer on Centre Court, if only to delay a post-match press conference in which her public spat with Maria Sharapova was quickly brought up.
The American insists the feud, which stems from a Rolling Stone magazine article, has not been a distraction, although she knows she needs to perform better than she did against Minella.
"I feel like I was a little rusty for some reason today," said Williams, who faces Spain's Caroline Garcia next.
"I don't feel like I played my best. I felt really upset when I lost my serve in the second set.
"With that being said, I think Mandy played really well.
"I thought she was really mixing up her shots, mixing up her game. It wasn't an easy match for me.
"I'm a little excited I was able to play a tough match and to get through it.
"After today, there's so many ways that I can improve and that I'm going to need to improve if I want to be in the second week of this tournament."
Williams beat Agnieszka Radwanska in last year's final and the Pole overcame Yvonne Meusberger 6-1 6-1 in the first round today, setting up a second-round tie with Mathilde Johansson.