Goals from Angel Di Maria, Gonzalo Higuain and Messi, with a brilliant free kick, gave Argentina their first home win over Paraguay in a World Cup qualifier in 39 years.
Argentina lead the South American group with 13 points from six matches, one more point than Chile and Ecuador and two ahead of Uruguay.
Paraguay, quarter-finalists in South Africa in 2010, remained one from bottom with four points before basement team Peru’s match at home to Venezuela.
Argentina went ahead in the second minute when Ezequiel Lavezzi crossed square into the middle, Higuain played the ball back to him and the winger passed to Di Maria who found the top corner with a sweet left-footed shot.
Paraguay got a lucky break and equalized in the 17th minute when holding midfielder Rodrigo Bran, playing for the suspended Javier Mascherano, gave away a penalty for handball.
Midfielder Jonathan Fabbro converted the penalty beating goalkeeper Sergio Romero with a good strike into the top corner from the spot.
Di Maria almost scored again direct from a corner which he took left-footed from the right, curling the ball in towards goal where goalkeeper Justo Villar just managed to punch the ball out.
Argentina regained the lead on the half hour when Higuain latched onto a defensive mistake on the right, fended off left back Richard Ortiz and shot low past Villar into the net off the inside of the far post.
Messi could have made it three shortly afterwards with a free kick from the left but his shot came back off the near post.
The Argentina captain should have scored early in the second half when dribbled through the defense and chipped over Villar but the ball hit the post.
Messi made amends with a brilliant free kick from more than 30 meters that dipped low inside the diving Villar’s left hand post in the 63rd minute to take his tally to 10 goals in Argentina’s last six matches.
Argentina had not beaten Paraguay in a World Cup qualifier at home since 1973. The five qualifiers between the two neighbors since then were drawn.
The defensive Paraguayans hardly troubled Romero but had the ball in the net at a late free kick although the effort was ruled offside.