Down 44-23, Nevada's Cody Fajardo had directed three touchdown drives in the fourth quarter as the Wolf Pack (3-3, 2-1 Mountain West Conference) wiped out a huge deficit to tie the score with 1:38 left on a 5-yard TD pass to Richy Turner.
Kaehler, who completed 23 of 32 passes for 286 yards and three 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 32 of 51 passes for 387 yards and two TDs, but had two fumbles that San Diego State (2-3, 1-0) turned into touchdowns in the third quarter.
Muema and Donnel Pumphrey each rushed for more than 100 yards for San Diego State. Muema had 134 yards rushing and two touchdowns and Pumphrey had 112 yards, including a spectacular 72 yard TD run in the first quarter.
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.
Brock scored his third touchdown with 4:33 left in the third quarter with a four-yard run that capped the Wolf Pack's third 75-yard drive of the night that made it 36-23.
Judge added another TD on a 5-yard pass from Kaehler that made it 44-23 before the comeback by the Wolf Pack.
The key play for Nevada in their fourth-quarter rally was San Diego State's decision to go on fourth and 5 from the Nevada 40 with 3:13 left with the Aztecs up 44-37.
Fajardo completed five passes, the last to Turner tying the score at 44 with 1:38 left and setting the stage for the Aztecs' overtime win.