Shot-shy Italy needed a scrambled second-half equaliser by Daniele De Rossi to snatch a 1-1 draw with Paraguay after the South Americans had looked like pulling off a shock victory in their World Cup opener Monday.
The Italians were slightly better in the second half, levelling after another goalkeeping blunder in the 63rd minute, but did precious little to change the pessimistic view of their fans and media who sent them to South Africa with a verdict of no chance.
Paraguay had beaten Brazil and Argentina in qualifying so had no need to be in fear of the holders but their dire World Cup record where they have not got past the second round in seven attempts, going out in the group stage four times, was always there in the background.
The game had meandered through almost an entire half of error-littered, unimaginative play before exploding into life in the 39th minute.
Aureliano Torres swung in a hopeful free kick from wide on the right but Alcaraz showed great determination to make it his, rising early and high above Fabio Cannavaro to power his header beyond Gianluigi Buffon.
It was a stunning blow for the Italians, who had looked completely untroubled until that point, albeit without ever looking remotely like scoring themselves.
Italy were forced to bring on Federico Marchetti for Buffon in goal at halftime but it was at the other end of the pitch where they needed inspiration.
Little was forthcoming but they levelled when Justo Villar flapped and missed a Simone Pepe corner and De Rossi stuck out a foot to steer the ball into the empty net from close range.
Either side of the equaliser Paraguay could have had a second as Enrique Vera and substitute Jonathan Santana shot wide when well placed.
Italy roused themselves in the last 10 minutes in search of a winner but stout defending saw Paraguay hold on for deserved point.
Italy, who face New Zealand and Slovakia in their remaining group games, will look to 1982 for inspiration, when they drew their opener but went on to win the tournament.