Jutkiewicz struck after Jason Shackell was harshly adjudged to have impeded Richard Keogh and Baker sealed the win in the closing stages to condemn County to a second straight defeat.
Derby forward Steve Davies flashed a shot across the face of Joe Murphy's goal in the opening minutes before Rams full-back Kevin Kilbane spurned a glorious opportunity to put Nigel Clough's side ahead on 12 minutes.
Ben Davies picked out the on-loan Hull man unmarked with an excellent outswinging corner from the right but Kilbane headed wastefully over from six yards out.
Jutkiewicz then had a free header at the other end from Sammy Clingan's corner but the Polish striker could only direct his header straight at Frank Fielding.
On 28 minutes Murphy produced a superb smothering stop to deny Jamie Ward after Clingan was caught in possession midway inside the Coventry half, before Martin Cranie blocked Lee Croft's shot minutes later.
The Sky Blues had failed to score in either of their previous games at the Ricoh this season and, in a fairly uneventful first half, rarely threatened Fielding's goal.
Left-back Chris Hussey hit a speculative long-range effort from the left that the County keeper comfortably caught but it was the visitors who looked most likely to make a breakthrough as the first half drew to a close.
Injured Coventry keeper Murphy was replaced by Chris Dunn at the interval and Dunn was immediately called into action to deny John Brayford's attempt from the edge of the area.
The hosts hit back straight away and finally began to show some attacking intent.
David Bell persevered inside the Derby area before pulling the ball back to top-scorer Jutkiewicz, who fluffed his initial attempt before Kilbane blocked his second effort on the line.
Jutkiewicz then turned Cyrus Christie's cross narrowly wide before the home side's early second-half pressure eventually paid off just before the hour mark, albeit in controversial fashion.
Clingan curled a corner in from the Coventry left and after Rams defender Shackell collided innocuously with Keogh, Jutkiewicz stepped up to send Fielding the wrong way from the spot.
Andy Thorn's men continued to press and substitute Roy O'Donovan stung the palms of Fielding before Baker put them in complete control with 11 minutes left.
Christie sent in a long-throw from the right-hand side and the Derby defence could only flick the ball on to midfielder Baker, who smashed home from close range.