Two policemen were among those hurt during a crush outside the Makhulong Stadium in Johannesburg, as the Nigerians faced North Korea.
Entry to the game was free, but many more than the 10,000 capacity turned up, prompting a crowd surge.
But Nigerian officials say they could not have organised another venue.
Nigeria Football Federation (NFF) board member Taiwo Ogunjobi told BBC Sport that it was impossible to find a bigger ground to host a game of such magnitude.
"We strived really hard to find a bigger stadium but unfortunately Fifa had taken control of all the stadiums from the South Africans," Ogunjobi said.
"Sadly this incident occurred but it was surely out of our hands because we and North Korea wanted to have this game.
"We had everything in place to have a successful match but unforeseen circumstances beyond our control took centre stage.
"Our sincere sympathies go out to the injured, but we are glad no lives were lost."
Several fans fell under a rush of people outside the ground in the Johannesburg suburb of Tembisa.
Football's world governing body Fifa has said it was not involved in any "operational" aspect of the fixture.
A Fifa statement added that "contrary to some media reports, Fifa had nothing to do with the ticketing of this game".
One fan, Japhta Mombelo, described the moment he was caught up in the chaos.
"The crowd overpowered me and I went down," he said. "I fell down and people just fell over me."
The first crowd surge came when the stadium gates were opened to allow fans entry.
Police soon closed the gates, but when they were re-opened, a second rush occurred, with more people falling over.
"When we were coming in, they were just stepping on us," another fan, Princess Mbali, who was wearing a South Africa shirt, said.
"I thought I was dying. I was at the bottom."
One policeman was seriously hurt after reportedly being crushed against a gate. He remains in a stable condition, according to the South African Police Services.
The crowd was mainly made up of local South African and travelling Nigerian fans.
Police said that hundreds of people were also holding photocopies of tickets and tried to break down the gates after the match had started and police had closed them.
In a separate incident the game was then stopped for five minutes in the second half when a railing collapsed inside the stadium.
Match commissioner Steve Goddard explained that he took the step to avert a possible stampede inside the ground.
"After consulting with match officials and the Nigerian contingent we agreed to stop the match, with the collapsed rail," Goddard said.
"It was only a matter of controlling the fans and we are glad disaster was averted."
The incidents have provided the wrong sort of build-up to the World Cup that the South African authorities would have wanted, the BBC's Piers Edwards, outside the stadium, says.
The venue is not a World Cup venue, but the incident is likely to heighten security even further ahead of the World Cup opener between South Africa and Mexico at Soccer City on Friday, our correspondent says.
The game itself finished 3-1 to Nigeria, with Yakubu Aiyegbeni, Victor Obinna Nsofor and Obafemi Martins scoring for the Super Eagles, while Jong Hyok Cha was on target for North Korea.