The defending champion dominated on her serve, winning the first set without dropping a point while serving. Her main weapon let her down only at the start of the second set, when Minella was able to take a 2-0 lead when Williams double-faulted on break point.
She was one point from going down 3-0 but then won 15 of the next 18 points to take a 4-2 lead, and broke again to secure the win.
"For me, it's the greatest moment for a tennis player to walk out on Centre Court," Williams said after her first match at Wimbledon since winning Olympic gold here last year. "That was such a great moment too. So many great memories on this court."
The top-seeded Williams has now won 32 consecutive matches, and is now 44-2 overall in 2013.
Williams moves on the play Caroline Garcia of France in the second round.
Top-seeded Novak Djokovic advanced to Wimbledon's second round with a 6-3, 7-5, 6-4 win over Germany's Florian Mayer on Tuesday.
Looking to win his second title at the All England Club, Djokovic was playing his first match of the year on grass after losing in the semifinals at the French Open. He managed the switch from clay easily, controlling most of the match and saving the three break points he faced.
Djokovic won his only Wimbledon title two years ago.
James Blake, routed Thiemo De Bakker of Netherlands 6-1, 6-3, 6-2, ending a five-match Wimbledon losing streak.
The 33-year-old American, a former top-five player currently ranked 87th, reached the second round at the All England Club for the first time since 2008. He lost his opening matches at the grass-court Grand Slam tournament each of the past four years.
Blake's best Wimbledon showings were reaching the third round in 2006 and 2007.
American teen Madison Keys earned her first win at the All England Club on Tuesday, beating Britian's Heather Watson 6-3, 7-5.
A rising star at 18, Keys doesn't have much experience on grass, but she showed potential with big serves and groundstrokes.
Also Tuesday, Kimiko Date-Krumm needed just 44 minutes to defeat German teen Carina Witthoeft 6-0, 6-2 in the first round.
The veteran Japanese player, at 42, is the second oldest player to have won a match at Wimbledon after Martina Navratilova, who was 47 when she reached the second round in 2004.
The 18-year-old Witthoeft was making her Grand Slam debut.
The former No. 4-ranked Date-Krumm reached the semifinals at Wimbledon in 1996, a few months after Witthoeft celebrated her first birthday.
Witthoeft struggled with her first serve and was broken six times by Date-Krumm, who came out of retirement in 2008.