The Barcelona striker gave Alejandro Sabella's side a 12th minute lead only for Jose Nunez to bring Paraguay level six minutes later courtesy of a fumble by Argentina goalkeeper Sergio Romero.
Manchester City striker Sergio Aguero then restored Argentina's advantage just after the half hour with a smart left-footed effort into the bottom corner.
Angel Di Maria's strike five minutes into the second half and another Messi penalty two minutes later then appeared to put the result beyond doubt.
Roque Santa Cruz gave Paraguay a late consolation on 86 minutes before a Maxi Rodriguez goal in injury time restored the visitors' three goal advantage.
The convincing result moves Argentina three points clear of Colombia at the top of the CONMEBOL qualifying group and assures them of a top-four place, meaning a spot at next year's World Cup is also now secure.
Colombia, meanwhile, missed their chance to seal a World Cup spot after losing 2-0 to Uruguay in Montevideo.
Jose Pekerman's side only needed a point to join Argentina in confirming a top-four place and they looked like achieving that aim for long periods of the match.
However, Edinson Cavani's 76th minute header edged Uruguay in front before Christian Stuani sealed the win four minutes later.
The defeat means Colombia will need a result from their final two games against Chile and Paraguay in order to book their place at Brazil 2014.
The three points for Uruguay lifts them into fifth place, which would secure a play-off spot against Jordan.
They are only separated from fourth-placed Ecuador on goal difference after Reinaldo Rueda's side could only draw 1-1 with bottom-placed Bolivia in La Paz.