Pereyra came close to opening the scoring in the first half, but the winger saw his shot come back off the underside of the bar.
However, it was he who grabbed to only goal of the game in the 56th minute, as his shot from the edge of the box beat Michael Agazzi.
Italy international striker Antonio Di Natale had the chance to put the game to bed when he found himself in an one-on-one situation with Agazzi, but the goalkeeper did well to thwart the 35-year-old.
Cagliari striker Mauricio Pinilla was sent off in stoppage time for a cynical foul on Giampiero Pinzi.
The win moved Udinese into fifth place in Serie A.