The Republic of Ireland Under-21 international captain fired home a first-half penalty to give the home side a deserved half-time lead, and doubled his tally at the end of a mad five-minute spell midway through the second period.
It began when Kari Arnason headed the Pilgrims further ahead on the hour to spark a period of three goals and two bookings in 300 seconds.
After Arnason's first Argyle goal, Reading immediately pulled it back to 2-1 as fellow Icelander Gylfi Sigurdsson headed his second goal in as many games.
Argyle responded by restoring their two-goal advantage straightaway when Judge headed home Chris Clark's cross and substitute Ashley Barnes added a late fourth.
Judge had opened the scoring after Jamie Mackie had won the Pilgrims a 13th-minute penalty. Mackie seized on a superb pass from Clark to beat Brynjar Ingimarsson to the ball and poke it past him.
The Reading captain was committed to the tackle and brought Mackie down as he skipped away from him, leaving Judge to step up and convert the penalty.
After two successive 1-0 home defeats, the goal was a tonic for the Pilgrims, who came close to doubling their lead when a sweeping passing move ended with Mackie sending a deft header narrowly wide.
Argyle went close again when Ingimarsson nearly turned the ball past his own goalkeeper, Adam Federici, after Karl Duguid had sent a low cross into the six-yard box.
Reading posed little real first-half threat to the Pilgrims goal until the final minute before the break, when Jobi McAnuff drove into the heart of the home side's defence.
However, goalkeeper Romain Larrieu was quickly down to turn the ball around his post.
After the break, Argyle continued to press, and Rory Fallon teed up Clark for a left-foot drive that smashed against the outside of the post.
Then came the three-goal five-minute mayhem that settled the game, with Barnes adding a late fourth to round off a fine afternoon for the Pilgrims.