Cade Cowell also scored for San Jose and JT Marcinkowski finished with three saves for the Earthquakes, who picked up a point for just the second time in their last eight matches (0-6-2).
Ramon Abila and Brent Kallman scored goals for Minnesota, which extended its unbeaten streak to seven games (4-0-3). Tyler Miller finished with one save.
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San Jose, outscored 15-2 over its previous seven games, jumped out to a 1-0 lead in the 15th minute on Cowell's team-high fourth goal of the season and second in two games.
The score was set up by Cristian Espinoza who stole Bakaye Dibassy's pass back attempt for Miller outside the right corner of the 18-yard box and then hit the 17-year-old Cowell with a crossing pass. Cowell then put a right-footed shot inside the left post.
The Loons tied it in the fourth minute of first-half stoppage time when Kallman, who entered in the 29th minute when Minnesota captain Michael Boxall left with a leg injury, scooped up a bouncing rebound on the left side of the six-yard box and then quickly fired a tight-angle left-footed shot around defender Tanner Beason and past Marcinkowski.
Minnesota United dominated action to start the second half and had a handful of scoring chances, including a shot by Franco Fragapane that bounced straight back out off the left post.
The Loons then took a 2-1 lead in the 69th minute after a VAR review determined Judson had dragged down Chase Gasper in the box for a penalty. Marcinkowski stopped Abila's penalty kick try but Abila followed up his shot and chipped in the rebound into the right corner of the goal.
Kikanovic then tied it, 2-2, in the 82nd minute, weaving his way through a pair of Minnesota defenders of the left side of the box and then firing a right-footed over Miller from about 10-yards out.
Minnesota's Hassani Dotson proposes after match
Minnesota United's Hassani Dotson gets down on one knee to propose to his girlfriend Petra Vuckovic, who excitedly says yes.
Benjamin Kikanovic's first MLS goal is the equalizer for San Jose
Major League Soccer: Benjamin Kikanovic (82') Minnesota United FC 2-2 San Jose Earthquakes
17-year-old phenom Cade Cowell scores 2nd goal in as many games
Major League Soccer: Cade Cowell (15') Minnesota United FC 0-1 San Jose Earthquakes