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In an unpredictable game at the Sardegna Arena, Kalinic scored two poacher's finishes in the first half to cancel out Joao Pedro's exquisite opener, and send the Giallorossi into the break ahead.
A cool finish from Justin Kluivert - his second in the space of a few days - then seemed to put Paulo Fonseca's side on course for victory, only for substitute Gaston Pereiro to give his side a lifeline with an emphatic finish.
For the second time in the game, however, Roma responded almost immediately: as Aleksandar Kolarov's inswinging free-kick from out wide evaded everything except the top of Henrikh Mkhitaryan's head and the back of the net.
And that would prove the winner, despite Joao Pedro setting up a tense finish as he converted the rebound after Pau Lopez saved his initial penalty.
Games at the Sardegna Arena regularly throw up dramatic moments for the Giallorossi and this game was no different, as a seven-goal thriller played out in enthralling fashion.
But Kalinic's brace gave the visitors the upper-hand in the contest before the break - an advantage they would always retain in some form, even if Cagliari threatened on occasions.
Robin Olsen was the busier of the two goalkeepers and made a number of incredible saves even before Joao Pedro's opener, but the creativity of Henrikh Mkhitaryan and pace of Kluivert and Cengiz Under regularly had Rolando Maran's side worried.
In the end the victory was secured by the tightest of margins - Kolarov's free-kick nestling home from an acute angle, with Mkhitaryan insisting he got a touch - as fans sweated through the six minutes of added time.
The Giallorossi are back in action next Sunday, as they host Sampdoria at the Stadio Olimpico.
They then travel to Seville for the first leg of the Europa League last-16 the following Thursday.
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