With in-form Cristiano Ronaldo rested by manager Maurizio Sarri, Argentina international Dybala provided the firepower the champions needed from a first-half free-kick after Brescia forward Florian Aye was sent off.
Sarri's men were guilty of wasteful finishing against their 19th-placed visitors, who might have ground out a point towards their battle against relegation had they kept 11 men on the field, but Cuadrado kept his cool to double their lead after 75 minutes.
It was far from a vintage performance, but two goals proved enough to earn Juve a first win in three matches and relieve some of the pressure mounting around Sarri's position.
Giorgio Chiellini's appearance as a second-half substitute following a long injury lay-off also boosted the home crowd.
An early clash with Gonzalo Higuain left Brescia goalkeeper Enrico Alfonso unable to continue and he was replaced by the club's third-choice stopper Lorenzo Andrenacci after nine minutes.
But Brescia's impressive first half fell apart 10 minutes before the interval.
Aye brought down Aaron Ramsey on the edge of the box and received his second yellow card in four minutes, before Dybala curled a sublime free-kick into the left corner of the net with Andrenacci rooted to the spot.
Andrenacci kept the deficit to one goal at the break with a superb one-handed save, turning a Daniele Rugani header onto the crossbar when the Juve defender looked certain to score.
Leonardo Bonucci was denied early in the second half, Andrenacci parrying the Italy international's powerful header away from danger.
Dybala danced through the Brescia box after 66 minutes only to have his shot from 12 yards blocked by Birkir Bjarnason.
Bjarnason blazed over the crossbar when the ball fell to him unmarked on the edge of the box moments before Cuadrado made sure of the victory, playing a neat one-two with Blaise Matuidi and applying a cool, dinked finish past Andrenacci from six yards out.
Higuain saw a header disallowed for offside late on and Dybala hit the bar right at the death, with Juventus comfortable in the end.