Thirty-six-year-old Antonio Di Natale, who back-tracked on last season's decision to retire, was as sharp as ever and scored both goals in Udinese's 2-0 win over promoted Empoli to follow his four-goal haul in the Coppa Italia last week.
AC Milan gave Filippo Inzaghi a winning start as coach by beating Lazio 3-1, helped by an inspired debut from goalkeeper Diego Lopez who capped his performance by saving a penalty in stoppage time.
Sampdoria trailed for 83 minutes at promoted Palermo and played the entire second half with 10 men after Vasco Regini was sent off, but still managed to draw 1-1 thanks to a goal by Daniele Gastaldello in the last minute.
Promoted Cesena beat Parma 1-0 and Sassuolo drew 1-1 with Calgiari in Sunday's matches.
Champions Juventus made a winning start to the season on Saturday with a comfortable 1-0 victory at Chievo Verona while title hopefuls AS Roma beat Fiorentina 2-0.
Napoli took an early lead at Genoa through Jose Callejon but the hosts levelled just before halftime with a superb header by Chile forward Mauricio Pinilla on his debut following his move from Cagliari.
Napoli, who failed to qualify for the Champions League after losing 3-1 to Athletic Bilbao due to a shambolic defensive display, grabbed the three points when Dutchman De Guzman sneaked in behind the Genoa defence at a free kick to score from close range.
Rafael Benitez's side were plagued by inconsistency last season and seem to be in for an equally turbulent ride this time around.
"Italian football needs to calm down and not consider every match a life or death situation," Benitez told reporters. "Before our goal people acted as if the title was beyond us and we were going into Serie B.
Inzaghi, the latest occupant in the Milan hot seat after Massimiliano Allegri and Clarence Seedorf were both fired last season, watched his side race to a 3-0 lead in just over an hour with goals from Keisuke Honda, Sulley Muntari and a Jeremy Menez penalty.
But Lazio, themselves playing their first competitive game under new coach Stefano Poli, suddenly came back in the game when Alex turned Candreva's cross beyond Lopez three minutes later.
Stefano Mauri clipped the crossbar as they continued to press, then Lopez made a superb close range save to stop Candreva before saving the same player's weak penalty in stoppage time.
Vidic conceded a first-half penalty with a foul on Fabio Quagliarella against Torino but his blushes were spared when Samir Handanovic saved Marcelo Larrondo's spot-kick.
The former Manchester United defender was given a straight red card in stoppage time, however, for sarcastically applauding the referee.
Inter coach Walter Mazzarri was furious with the referee, claiming his side should have had a penalty for a foul on Mauro Icardi.
"If the referee gives the first penalty, he should have given the one on Icardi, which was much clearer than the one on Quagliarella," he told reporters.
"As they do in England, Vidic applauded a decision he made and the referee interpreted it wrongly. It was an absurd red."
Di Natale, Serie A top scorer in 2009/10 and 2010/11, opened his account in the 57th minute after a playing a one-two with Luis Muriel, then pounced on a bad back pass for his second goal five minutes later