The South American side looked on course to beat the World Champions until Villar completely missed a Simone Pepe corner which allowed De Rossi to tap in from inside the six-yard box.
And after earlier goalkeeping mistakes from England's Robert Green and Algeria's Faouzi Chaouchi, South Africa 2010 has so far been memorable for the scale of goalkeeping errors on display.
The appalling weather conditions adversely affected the level of ambition of both sides as neither hit the heights during the early stages.
In fact, the match was terribly pedestrian as both outfits seemed reluctant to really get forward.
Pepe and Alberto Gilardino tried to occasionally push and probe for Marcello Lippi's side but they were continually rebutted by a disciplined Paraguayan backline that frequently harried and closed down Italy.
Yet, in reality, Paraguay were fortunate to even have 11 men on the field following a terribly rash tackle by Cristian Riveros after just 60 seconds.
He clattered into Riccardo Montolivo and could easily have broken his leg and, although it warranted a red card, he was saved by the fact the tackle was made so early on in the contest.
Aureliano Torres should perhaps have done better for Paraguay after 21 minutes but his shot was not strong enough and, although Gilardino had been Italy's biggest hope for a goal prior to the encounter, he was virtually anonymous during the first half.
As half-time approached, and just as the opening period looked destined to end goalless, Alcaraz changed all that with a brilliant header to shock the World Cup holders.
He climbed above Fabio Cannavaro and De Rossi to head home Torres' drifted free-kick.
After the interval, Paraguay continued to work hard at closing down the Italians but they were then spectacularly let down by Villar.
Pepe's corner was simple enough to save but he virtually dived out of the path of the ball and De Rossi could not miss to grab the equaliser.
The goal was exactly the motivation Italy needed to try and go on to win the match and their superior quality did start to shine through as the end approached.
Montolivo pushed further and further forward while Antonio Di Natale also worked hard after replacing Gilardino.
However, despite totally dominating the last half an hour of the match, Italy could not find a second to cap off what would have been hard-fought comeback.
Italy: Buffon (Marchetti, 46), Cannavaro, Zambrotta, Chiellini, De Rossi, Montolivo, Pepe, Criscito, Marchisio (Camoranesi, 59), Gilardino (Di Natale, 72), Iaquinta. Subs (not used): De Sanctis, Palombo, Pirlo, Maggio, Quagliarelli, Gattuso, Pazzini, Bonucci, Bocchetti.
Paraguay: Villar, Da Silva, Torres (Santana, 60), Alcarez, Bonet, Morel, Riveros, Vera, V Caceres, Valdez (Santa Cruz., 68), Barrios (Cardozo, 76). Subs (not used): Bobadillia, Barreto, Caniza, Veron, Barreto, Benitez, J. Caceres, Ortigoza, Gamarra.