Hamsik had earlier set up Cavani's leveling header with 17 minutes to play.
The Reds survived as a Marek Hamsik shot was cleared off the line by Paul Konchesky but they had chances to win.
Weiss's trickery on the left set up chances for Hamsik, who bent an ambitious shot wide, and Stanislav Sestak, who fired narrowly wide.
Marek Hamsik's opener was cancelled out by an own-goal from Paolo Cannavaro, but Cavani popped up with five minutes remaining to seal the win.
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Paston partly redeemed himself with a smart tip-over from Hamsik's long-range effort shortly before half-time, but within five minutes of the restart, he was beaten.
Seconds later Marek Hamsik toe-poked forward for Vittek, who showed none of the composure he demonstrated to score twice against Italy by firing straight at Stekelenburg.
Earlier Sunday, Edinson Cavani took his Serie A-leading goal tally to 28 as second-placed Napoli came from behind to beat Siena 2-1 thanks to Marek Hamsik's injury-time winner.
Slovakia's main threat in the first half, such as it was, was provided by 20-year-old Manchester City winger Vladimir Weiss, as the much-vaunted Napoli playmaker Marek Hamsik struggled to impose himself.
Hamsik snatched the winner 10 minutes from time from Datolo's cross to give coach Walter Mazzarri a record of three wins in four league games since replacing the sacked Roberto Donadoni.
Edison Cavani outpaced Carragher and crossed for Hamsik, who turned in a shot at goal which was cleared from the Reds goalmouth by Konchesky and the official behind the goal ruled the ball had not crossed the goal-line.
Sebastian Giovinco doubled the lead in the 54th minute after a mistake by Napoli defender Matteo Contini, but Marek Hamsik pulled a goal back five minutes later with his seventh of the season from Jesus Datolo's low cross.
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