Vydra took his tally for the season to 12 in all competitions as he followed up a slick first-half strike with a second-half penalty and another smooth finish after an error from Darren Randolph.
The Czech striker was too hot to handle all afternoon, and his treble sees an in-form Derby rise to fifth position after a run of six wins in their last eight games.
Middlesbrough suffered an afternoon to forget, with Daniel Ayala’s second-half dismissal for a second yellow card summing up their struggles.
Boro went into the game on the back of a comfortable home win over Birmingham three days earlier, and were the brighter side in the opening stages. Martin Braithwaite drilled a long-range free-kick narrowly over the bar, before Boro almost claimed the lead through the in-form Britt Assombalonga.
Assombalonga already boasts 11 goals for the season, and he almost added a 12th when he stabbed in a shot after Ayala flicked on Cyrus Christie’s long throw, only for Derby goalkeeper Scott Carson to make a superb point-blank save.
The Rams had barely figured as an attacking threat to that point, but Gary Rowett’s side took the lead with their first meaningful foray forward less than a minute later. It was a well-worked goal, with Vydra drifting into the area and playing a slick one-two with former Boro forward David Nugent before lashing a fierce 12-yard finish past Randolph.
Derby went close again when Tom Lawrence cut in from the left-hand side to drill a low shot straight at Randolph, but Boro finished the first half strongly and created two decent opportunities.
Assombalonga headed Stewart Downing’s corner wide of the left-hand post, before full-back Fabio Da Silva also failed to find the target as he nodded Cyrus Christie’s cross over the bar from inside the six-yard box.
They proved to be crucial misses, as Derby doubled their lead from the penalty spot in the first three minutes of the second half. Braithwaite conceded possession in his own half, enabling Nugent to play Vydra into the area. The Czech forward went tumbling under a challenge from Ayala, and picked himself up to roll the subsequent spot-kick into the corner of the net.
Ayala could have been sent off for the foul, given he had already been booked in the first half, but it did not take him long to book himself an early bath anyway. He senselessly clattered into the back of Nugent as the Derby striker held up the ball close to the halfway line, and referee Simon Hooper rightly issued a red card.
With a man advantage, it only took Derby three minutes to extend their lead, with Vydra claiming his hat-trick. Ben Gibson’s back-pass put Randolph under pressure, and with the Boro goalkeeper only able to spoon the ball up in the air, Vydra gratefully received possession before slotting home.