Hirscher had fallen 107 points behind Svindal in the overall standings, but the Norwegian sustained ruptured knee ligaments in a crash in Saturday's downhill, ruling him out for at least nine months.
That has opened the door for Hirscher, who finished joint fourth in the first leg of the slalom, 0.34sec off German Fritz Dopfer's leading time of 50.14sec, to claw back some points with an eye on claiming an unprecedented fifth consecutive World Cup overall title.
Norway's Henrik Kristoffersen, winner of four slaloms so far this season, was in 12th spot at 0.83sec.
After the spectacular downhill, which organisers brought to a close after just 30 runners following crashes by not only Svindal but also Austrian duo Hannes Reichelt and Georg Streitberger, the slalom got off to the worst possible start.
Italian Giuliano Razzoli was first out of the hut, but the 2010 Olympic slalom gold medallist fell heavily and was left clutching his left leg.
The helicopter was called up, as it had been during the downhill, to evacuate Razzoli before competition restarted on the testing, icy Ganslern piste.
Also skiing out was the French winner of Friday's combined event, Alexis Pinturault.
But it will be a loaded second run, scheduled for 1230 GMT, with German Felix Neureuther, Frenchman Jean-Baptiste Grange, Russian Alexander Khoroshilov and Swedes Mattias Hargin, Anton Lahdenperae and Andre Myhrer all still in the mix along with Hirscher's teammates Marco Schwarz and Manuel Feller.