Nigeria were lucky to escape with such a narrow defeat after being outplayed by Argentina, who spurned several clear-cut goalscoring opportunities despite not living up to expectations in their Group B opener.
Diego Maradona's men secured all three points via a header from left-back Gabriel Heinze after six minutes.
The 32-year-old, a pivotal member of Marseille's French Ligue 1 winning squad last season, met Juan Veron's corner with a diving header that found its way past Nigeria keeper Vincent Enyeama, who saved his side from an embarrassing loss.
"Unfortunately, there was a mistake from our side that created the corner kick," Swedish coach Lagerback, who took over at Nigeria four months before the World Cup, said after the game.
"We were sleeping on one of the corners and that was enough. We were punished."
Eneyama had a night to remember although Nigeria failed to show any signs that they can make it through to the second round for the third time in their history, having achieved that target in 1994 and 1998.
He pulled off a string of impressive saves to thwart the likes of Lionel Messi and Gonzalo Higuain.
The 27-year-old guardian, who plays for Israeli side Hapoel Tel Aviv, believes Nigeria can make amends.
"We're not going to kill ourselves. We have two more games to play," he said.
"We thank God the score was a little encouraging."
Nigeria next face Greece on Thursday before wrapping up their group stage campaign against South Korea on June 22.