The result left the Serbians top of the standings with 15 points from seven games, two ahead of Ireland who lost ground after being held to a 1-1 draw in Georgia.
Wales and Austria, who meet in Cardiff later on Saturday, have eight points each.
Serbia will head confidently to Dublin for Tuesday's clash with the Irish after a fine display in which the 22-year-old Gacinovic stood out, having scored on his full debut as a substitute in a 3-1 win at Georgia in March.
Coach Slavoljub Muslin's preferred 3-4-3 formation pinned the Moldovans back from the start and Gacinovic drilled a 12th-minute shot into the top corner after a sharp low cross by Aleksandar Kolarov.
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The pair swapped roles on the half-hour as Gacinovic sent a fizzing low cross from the right and Kolarov, deployed as a left winger swept the ball home from seven metres.
Striker Aleksandar Mitrovic shaved the outside of the post with a close-range header as some Serbian fans ran up the terraces for cover from bursts of driving rain.
Serbia continued to dominate in the second half and Mitrovic put the icing on the cake with a predator's finish after captain Branislav Ivanovic and substitute Adem Ljajic had gone close.
The Newcastle United striker slid in to bury a Filip Kostic cross in the 81st minute as Serbia stayed in the driving seat to reach their first major tournament since the 2010 World Cup in South Africa.
(Editing by Ed Osmond)
