The capital club approached Sunday night's match searching for their first win in six games in all competitions, at the end of a week which saw president Claudio Lotito turn up to training unexpectedly to deliver a motivational speech.
Petkovic knew his boss would not visit him again without bringing worse tidings so, after watching Cagliari dominate the first half without scoring, he threw Klose into the fray to spark a revival.
The former Bayern Munich star did not disappoint, heading home the pressure-relieving opener on 52 minutes before Antonio Candreva's penalty gave Lazio a comfortable cushion heading into the last half hour.
The win boosted the Romans into seventh place while a third away defeat of the season left Cagliari in 13th.
Two of Lazio's last three games had ended goalless to seriously frustrate their fans and, in light of defeat at Atalanta last time out, goals were sorely needed to win them over.
The first half would instead end with boos and whistles towards Lazio's retreating players, with Cagliari unfortunate not to take a lead down the tunnel at the break.
Marco Sau was key to the Sardinians' game plan and, after failing to properly test Federico Marchetti with a tame shot, he fed Matias Cabrera for an effort that was deflected to safety.
Marchetti had to move quickly to smother the ball at Davide Astori's feet and the Lazio goalkeeper had Diego Novaretti to thank when he shepherded Victor Ibarbo out of a shooting position.
The jeers from the north stand abated briefly when Brayan Perea nodded Candreva's cross just over but they returned even louder when Lorik Cana managed to repeat the trick from a better position.
Both sets of supporters had cause to feel aggrieved at the half-time whistle, yet Petkovic bore the brunt of the local hostility.
It was time for the Bosnian tactician to gamble and a double substitution saw the talismanic Klose take to the Olimpico pitch for the second half.
The German hitman scored the opening goal with practically his first touch.
Petkovic may have showed little emotion on the touchline as Klose streaked away in celebration after heading Candreva's cross into the top corner but the goal fully energised his players.
The second arrived from the penalty spot in the 55th minute, Candreva slamming the ball high into Michael Agazzi's net following a handball by Francesco Pisano.
Perea fancied the third goal and nearly found it after being served by Ederson.
He was all alone in front of the keeper when his header clattered a post, with Klose failing to tap the loose ball into an unguarded net.
The zip had gone out of Cagliari's challenge but on another night Andrea Cossu's free-kick might have bobbled into the net with a heavier deflection off Astori.
There was, however, no deserved consolation goal.