After a tight first half in which both sides took a cautious approach, understandably given the stakes, Cagliari emerged with more conviction and Sau struck twice early in the second half to earn the points.
Pescara ended the game with 10 men with Vladimir Weiss sent off in stoppage time as the home fans made their feelings heard.
The win takes them eight points clear of their hosts, who occupy the first relegation berth, four points adrift of safety.
Both sides were in danger coming into the game and the early exchanges reflected this.
Pescara could ill-afford to lose another game at home while Cagliari knew a point would be worth its weight in gold against a rival for the drop.
Chances were at a premium with both defences unwilling to let their opponents get too close to goal.
Unsurprisingly, the only threats came from long range with Albin Ekdal missing the target by a few yards from fully 40 yards in the sixth minute.
Gaetano D'Agostino provided Pescara's response with a 30-yard effort which was a bit closer to the target, but not close enough.
Sau gave Cagliari the lead in the 54th minute when he met Andrea Cossu's cross with a header inside Mattia Perin's left-hand post.
Pescara's heads dropped and they soon conceded a second as their resistance broke completely.
Francesco Pisano sent in the cross from the right which was collected by Sau, who turned and shot into the same corner as his first goal to double Cagliari's advantage.
The home fans vented their frustration at their team's efforts, and Weiss replicated with some frustration of his own as he received a straight red card in stoppage time for an unnecessary foul on Victor Ibarbo.