Pereyra, who hit the crossbar in the first half, made amends with a more difficult effort midway through the second half to lift Udinese into a qualifying berth for the Europa League, ahead of the remainder of the weekend's matches tomorrow.
Cagliari's reaction was too little too late and they ended the game with 10 men after Mauricio Pinilla was shown a red card deep in stoppage time.
The game was officially a home one for Cagliari, despite being staged in the distant Trieste.
In fact, given the location just 50 miles from Udine, there were more visiting fans in the Stadio Nereo Rocco, and Udinese made it look more like a home game on the field.
Pereyra should have given them the lead in the 11th minute, but he struck the crossbar with a shot from just six yards out.
It was the biggest chance in a bland first half which saw Antonio Di Natale also test Cagliari goalkeeper Michael Agazzi from distance.
Cagliari produced nothing in attack and deservedly fell behind in the 66th minute.
Pereyra made up for his easy miss in the first half by taking a more difficult opportunity from the edge of the penalty area, curling his shot low into the far corner of Agazzi's goal.
Ten minutes later, Di Natale missed a good chance to seal victory, but after beating the offside trap, he touched the ball too far in front of himself, allowing Agazzi to collect.
Udinese's lead never looked in any danger, despite a late rally from Cagliari which still failed to produce any shots on target.
And the hosts lost Pinilla deep into stoppage time when he let his frustration get the better of him and charged Giampiero Pinzi from behind near the touchline.