They continued to push forward and after being repeatedly thwarted by the feet of Metz goalkeeper Alexandre Oukidja, they got the goal they craved when Draxler headed into an empty net after Oukidja parried a stinging cross from Angel di Maria.
But PSG, beaten Champions League finalists last month, continued to search for a first goal of the season and finally got one thanks to the determination of Angel di Maria and the opportunism of Draxler.
Di Maria trapped a long ball down the left wing and cut inside, producing a stinging cross which keeper Oukidja spilled into the air.
German Draxler pounced to head into the unguarded net, providing much relief to the home fans who were scattered across the Parc des Princes due to social distancing measures.
"We missed so many great chances but the players showed a great mentality and they ran like crazy. We will savour this match for a long time," Thomas Tuchel said.
Moments before Draxler's goal, Metz came agonisingly close to finding an unlikely winning goal when Youssef Maziz broke into the area but side-footed wide of the target.
"From the start we knew this was going to be a very difficult match right until the end," PSG defender Marquinhos said.
"We are still recovering and we wanted to break them down long before we did but it was so important to win, especially as we were at home and after losing the first two games."