Dinamo went on to beat SK Tirana of Albania 3-0, the joint biggest win of the opening night of action in the first legs of the first qualifying round, along with Vardar Skopje's 3-0 win over Welsh champions Barry Town in Macedonia.
Melkadze had a dream debut for his new side, scoring after just six minutes for Dinamo, who have probably done enough to reach the second qualifying round and a match against Austrians Graz AK after the return leg next week.
But there were no Dinamo fans there to see the victory as the game was played behind closed doors on UEFA orders following bottle-throwing incidents during Dinamo's UEFA Cup first round match against Slovan Liberec last season.
Georgi Anchabadze and Vitali Daraselia (penalty) added the others, while Tirana's Indrit Fortuzi missed a penalty after 60 minutes.
The other big winners on a night of only 12 goals in the 10 matches played were Vardar Skopje, who beat Barry 3-0 and now seem set for a tough second qualifying round tie against CSKA Moscow.
Wandair Dos Santos scored twice for Vardar after 38 and 69 minutes with Dejan Ristovski scoring the other three minutes after half-time.
Sliema Wanderers scored a well-merited European victory in searing 37 degree Celsius temperatures in Malta and gave themselves a chance of a place in the second qualifying round with a 2-0 win over Latvian champions Skonto Riga.
Sliema's two goals came from debutant Daniel Bogdanovic and Danilo Doncic (penalty) in a four-minute spell which followed the dismissal of Riga's Aleksandrs Isakovs, who picked up his second red card for a foul on Bogdanovic. The eventual winners will play FC Copenhagen.
Kaunas of Lithuania were the only away team to score and the goal was enough to give them victory over HB Torshavn in the Faroe Islands.
Martin Opic found the net after 75 minutes but Andreus Gedgaudas of Kaunas will miss the second leg after being sent off seven minutes from time.
Marco Haber of Omonia Nicosia was also sent-off in his team's 0-0 draw with Irtysh Pavlodar of Kazakhstan in Cyprus and he will also miss the second leg next week.
Some major teams from leading European soccer nations enter the competition in the third qualifying round next month, while the elite clubs take part from the group phase which starts on September 16.
