Hulse could have had a hat-trick but his late penalty was well saved by Dimi Konstantopoulos.
Coventry finished the game with 10 men after Jordan Clarke was sent off just after the hour mark.
Having conceded a goal only 28 seconds into their home defeat to Reading last weekend, Coventry turned the tables at Pride Park by taking the lead within five minutes.
Freddy Eastwood's left-wing cross was headed goalwards by Best. With players in front of him, Rams keeper Stephen Bywater opted to punch the ball but his clearance went straight back to Best who fired a volley into the bottom corner from 15 yards.
Derby responded well to the early setback and created two chances in as many minutes.
Coventry failed to clear Gary Teale's ninth-minute corner and the ball fell to Dean Leacock, whose shot looked like it would nestle in the bottom right-hand corner before it was deflected wide by Best on the six-yard line.
Hulse's turn and shot then skidded narrowly past the same post.
Derby felt they should have been awarded a penalty when Teale's far-post corner was headed across goal by Leacock to fellow centre-half Shaun Barker, who was upended by Elliott Ward's sliding tackle as he shaped to shoot.
There was an off-the-ball tussle between fiery Derby striker Paul Dickov and Coventry's debutant defender Leon Barnett in the 34th minute. The incident resulted in both players receiving a yellow card.
Dickov then spurned a gilt-edged chance to equalise in the last of five added minutes at the end of the first half.
Stephen Pearson surged into the box and sent a low ball across the face of goal. Dickov looked certain to score as he arrived on the six-yard line but the on-loan Leicester player somehow blasted his shot over the crossbar.
However, the diminutive Scot made amends after the break providing the assists for both of Hulse's goals.
His diagonal cross into the box four minutes after the restart was brilliantly cushioned on the chest by Hulse who then flicked the falling ball beyond Konstantopoulos.
Dickov then retrieved Pearson's cross to tee up his strike partner for a crisp first-time finish after 62 minutes.
In between, Teale sent a dangerous shot across the face of Coventry's goal after backing Clarke into the box.
It would be Clarke's final contribution as he was sent off a minute after Derby's second goal after receiving a second yellow card for bringing down Robbie Savage.
The defender had been booked in the first half for mis-timing a challenge on Dean Moxey.
Derby had the opportunity to ensure all three points with 10 minutes remaining when Chris Hussey, realising he was unable to intercept Teale's cross, nudged Lee Croft in the back leaving the referee with no choice but to award a penalty.
However, Konstantopoulos got across to the bottom right-hand corner to save Hulse's kick.