Down 44-23, Nevada's quarterback Cody Fajardo had directed three touchdown drives in the fourth quarter as the Wolf Pack wiped out a huge deficit to tie the game with 1:38 left on a 5-yard TD pass to Richy Turner.
Kaehler, who completed 23 of 30 passes for 286 yards and two touchdowns, threw a 9-yard pass to Adam Muema on a first-down play in overtime.
Three plays later, Kaehler found Judge over the middle from 13 yards.
The Aztecs defense stifled Fajardo on three passes after a motion penalty put the Wolf Pack back to the 30.
Fajardo completed 33 of 52 passes for 393 yards, but had two fumbles that San Diego State turned into two touchdowns.
Muema and Donnel Pumphrey each rushed for more than 100 yards for San Diego State.
The Aztecs, the defending Mountain West Conference co-champions, are 2-3, 1-0 in the conference while Nevada is 3-3 and 2-1 in the MWC.
Kendall Brock rushed for 107 yards and three touchdowns for Nevada.
Late in the second quarter, the Aztecs were trailing 17-14. They stunned the Wolf Pack first with a safety with 46 seconds left and then with a 25-yard touchdown pass from Kaehler to Dylan Denso with 9 seconds left for a 23-17 halftime lead.
The Aztecs came out in the third quarter and forced a fumble by Fajardo, which linebacker Josh Gavert picked up and returned 42 yards to the 9-yard line. On the first-down play, Muema scored his second TD of the night for a 29-17 Aztec lead.
A few minutes later, the Aztecs forced another Fajardo fumble, which they turned into another TD, a 5-yard run by Tim Vizzi with 8:09 left in the third quarter and it was 36-17.
Judge added another TD on a 5-yard pass from Kaehler that made it 44-23 before the comeback by the Wolf Pack.