Jose Nunez cancelled out Messi's early opener, but further goals from Sergio Aguero, Angel Di Maria and Messi again put Alejandro Sabella's Albicelestes firmly in control.
Roque Santa Cruz grabbed a second goal in the 86th minute, but Maxi Rodriguez swiftly restored the visitors' three-goal cushion in the final minute to ensure Argentina qualified for next year's tournament in Brazil in some style - and with two games to spare.
Barcelona forward Messi got the Argentinians off to the perfect start by converting a 12th-minute spot-kick after Aguero had been fouled by Paraguay goalkeeper Roberto Fernandez.
Paraguay equalised six minutes later when Nunez connected with a cross from the left - albeit with a little help from Argentina goalkeeper Sergio Romero, who failed to catch an easy ball.
The visitors restored their lead in the 31st minute through Manchester City forward Aguero, who produced a superb left-footed finish despite the close attention of two defenders.
Real Madrid's Di Maria made it 3-1 four minutes after the interval following a fine pass from Fernando Gago, and Messi despatched another spot-kick three minutes later to further widen the gap.
Santa Cruz reduced the deficit in the closing stages, but Maxi consolidated the win one minute from the end.
The triumph moved Argentina onto 29 points in the Conmebol qualifying zone - three above Colombia, who lost 2-0 against Uruguay in Montevideo earlier on Tuesday.
Di Maria has set his sights on finishing top of the standings, before - rather more ambitiously - lifting the cup in Brazil next year.
He told local Channel 7: "We have two games left and we hope to finish this competition as well as possible. If we are champions, it would be great.
"We have to go step by step. A great group was built by Alejandro (Sabella) and we are very happy."
Messi said on Fox Sports: "Qualification was the goal we've been looking for for a long time. Achieving that with two games remaining is very important.
"We have achieved our goal and now we are hoping to keep winning and progressing. We are on the right path but there is a long way to go yet in order to reach the World Cup tournament in the best form.
"The World Cup is seven games and you have to be in perfect form. You cannot make mistakes. We have already beaten big teams, but we still need to make some improvements to be world champions."