The Slovenian captain sent in a long-range shot that Algeria goalkeeper Fawzi Chaouchi misjudged and allowed to bounce into the net off his arm in the 79th minute.
“I was just lucky to see it go in like that,” Koren said.
Algerian coach Rabah Saadane said the new World Cup ball may have been a factor in the goal.
“Everyone saw what happened with the ball, and what happened yesterday with England's goalkeeper,” Saadane said. “You have to adjust to the flight of the ball.” Algeria substitute Abdelkader Ghezzal had been sent off seven minutes earlier, picking up a second yellow card for handling the ball inside the Slovenian penalty area.
It was the tiny Balkan nation's first ever World Cup victory after losing all three group matches at the 2002 tournament.
Algeria twice came close in the first half when Nadir Belhadj's 25-meter (yard) free kick forced Slovenia goalkeeper Samir Handanovic to stretch and push the ball over the crossbar, and defender Rafik Halliche headed just wide.
Both sides knew heading into the match that a loss would be costly, with difficult encounters against England and the United States left on the program. The Group C favorites drew 1-1 Saturday after a blunder by goalkeeper Robert Green cost England the victory.
Saadane said Chaouchi apologized to the team for fumbling the ball when Slovenia scored Sunday.
“We must recover and prepare for an even more difficult match against England, which is the best of the group,” Saadane said. “It's going to be very difficult now. We had a great opportunity today and we missed it.” The players in Polokwane showed nerves in the first half, frequently losing possession with inaccurate passes and crosses.
Slovenia's only scoring chance before the break was Valter Birsa's long-range attempt in the 43rd, tipped over the bar in a spectacular right-hand save by Chaouchi.
There was some commotion in the stands during the break when former France great Zinedine Zidane, who is of Algerian descent, was forced to climb to a higher section to get away from autograph-seeking Algerian fans who were pushed back briskly by security guards.
The Group C game at the newly built Peter Mokaba Stadium was the first ever World Cup match played on a surface partly consisting of artificial grass. Similar turf is being used at some major stadiums in Europe.
“I do not agree with this turf,” Slovenia coach Matjaz Kek said. “It's no excuse, it's the same for Algeria as well. But it is a different game on this surface.” The match could have taken a different turn two minutes before Koren's winner when Algeria midfielder Karim Ziani caught defender Marko Suler off guard and pushed toward the Slovenian goal but was cut off by Handanovic.
Algeria pressed for an equalizer in the final minutes but was unable to break through the Slovenian defense.
“It was a mental victory,” Kek said. “We wanted to win so much so it's great for our confidence to get that win.” Algeria beat North African rival Egypt in a tense playoff to qualify for the World Cup for the first time since 1986.
Slovenia reached its second World Cup with an upset playoff victory over Russia.
Slovenia faces the United States next while Algeria travels to Cape Town to take on England.
Fawzi Chaouchi, Madjid Bougherra, Nadir Belhadj, Anther Yahia, Rafik Halliche, Medhi Lacen, Rafik Djebbour (Abdelkader Ghezzal, 58), Karim Matmour (Rafik Saifi 81), Karim Ziani, Hassen Yebda, Fouad Kadir (Adlane Guedioura, 82).
Samir Handanovic, Miso Brecko, Marko Suler, Bostjan Cesar, Robert Koren, Valter Birsa (Nejc Pecnik, 84) Milivoje Novakovic, Bojan Jokic, Zlatko Dedic (Zlatan Ljubijankic, 53), Andraz Kirm, Aleksander Radosavljevic (Andrej Komac, 87).
