Srna shook off an ankle knock in the first half and handed Ukraine's champions the win over their stubborn Serbian rivals, whose congested midfield and desperate defending kept the 2009 UEFA Cup winners at bay until the 71st minute.
Shakhtar's Brazilian strikers Luiz Adriano and Jadson were denied several times by superb goalkeeping from Vladimir Stojkovic, but he then made a costly error which allowed Srna to break the deadlock.
The Serbia keeper parried long range efforts from Willian and Douglas Costa either side of a double close-range save from Adriano and Oleksiy Gai in the first half.
He thwarted the onrushing Jadson and tipped away an Adriano header shortly after the interval but then drifted too far to his left for Srna's free-kick, which whizzed past him at the near post.
"It was difficult to play against a team so quick and individually more skillful. We played as well as we could and were a bit unlucky with the goal," Partizan coach Aleksandar Stanojevic told Serbian television.
"We didn't want to play a defensive game but were forced on to our back foot by a more resourceful team and it will be difficult to win points in this group," he said.
He added: "We are looking forward to our next match at home to Arsenal, it will be a mountain to climb but it will also be a great opportunity for our fans to see a top team in action and for our young players to prove their worth."
STIFLED OUT
Partizan, who have won three Serbian league titles in a row and enjoyed a 17-match unbeaten run under Stanojevic in all competitions stretching back to last season, failed to muster a single shot on target in the first half.
The one-way traffic underlined how hard Partizan will have to work to adjust to a different level in Europe's premier club competition, which they also reached in 2003.
Their Brazilian hitman Cleo, who scored eight goals in six qualifying matches to steer them into the group stage, was not allowed even a sniff at goal. Even so, Partizan almost snatched a late draw against the run of play.
Shakhtar keeper Andriy Pyatov made a pair of last-gasp saves to deny midfielders Nemanja Tomic and Sasa Ilic, making amends for an earlier error when Mladen Krstajic's shot was cleared off the line by a defender.
The hosts were inches away from adding a second in injury time when Gai's scorcher from 18 metres hit the inside of the post and rebounded across the face of goal.
