Pasquale Foggia gave the hosts the lead on the hour mark but Franz Schiemer equalised in the 82nd minute before Janko struck seconds from time to hand the Austrian champions all three points in Rome.
The visitors had the first chance of the game in the 20th minute with Christoph Leitgeb's left-footed strike from the edge of the area met by a fine save from Lazio goalkeeper Albano Bizzarri.
Bizzarri then did well to anticipate Romen Tchoyi, who was ready to fire after a fast counter-attack by the Austrians.
The hosts finally put a good move together in the 26th minute with Mauro Zarate playing in Julio Cruz, whose left-footed strike from the edge of the area forced Salzburg goalkeeper Eddie Gustafsson to fully stretch to clear.
On the half-hour mark, Lazio almost shot themselves in the foot with Emilson Cribari's clearance from Dusan Svento's cross forcing Bizzarri to make a save.
The visitors demanded a penalty shortly before half-time after Janko fell inside the area with Roberto Baronio close to him but the referee played on.
Lazio almost took the lead early in the second half with Foggia's diagonal effort from outside the area going inches wide of the far post.
Zarate then tried his luck from distance but his central effort was saved by Gustafsson.
Foggia then fed Zarate inside the area whose right-footed shot went wide of the near post as Lazio got closer.
Foggia finally broke the deadlock shortly after.
The Italian midfielder's 30-yard strike towards the far post beat Gustafsson.
The home fans' joy did not last long, however.
The visitors pressed forward and their efforts paid off.
Bizzarri failed to clear substitute Admir Vladavic's cross from the right leaving Schiemer to fire from close range into the back of the net.
Lazio made another mistake with seconds left to play.
A poor head clearance by Cribari inside his own area proved an assist for Janko, who fired past Bizzarri.