England made the quicker start with new caps Paul Konchesky and Jermaine Jenas looking in confident mood.
However, the visitors' dominance failed to bring an early goal.
In fact, Slovenia nearly took the lead as Aston Villa defender Gareth Barry played a back-pass to Leeds goalkeeper Paul Robinson, only for the ball to go over his head and just bounce over the bar.
Slovenia also created a chance early on but Krsic's curling free-kick was handled confidently by Paul Robinson.
A minute later, David Dunn's 30-yard cross at goal caught the Slovenian goalkeeper Saso Fornezzi out, but the ball ended up on the roof of the net.
As the home side took the match to England, a Konchesky free-kick found the head of Chris Riggott, but his effort looped over the bar.
Dunn came close to scoring again as the skipper let go a low strike that found the wrong side of the Slovenian net.
With ten remaining in the half, England made a break-through when West Ham's Jermaine Defoe went clear in on goal, only for Slovenia's Mathias Lunder to bring him down in the box. The referee had no choice but to send him off.
Dunn stepped up to take the penalty, but he sent his shot horribly over the bar.
England got one more chance before the break through Liverpool's Stephen Wright's cross, which found Charlton's Scott Parker, but his header found its way into the keeper's arms.
The breakthrough came in the second half.
Just after the hour mark, Derby's Malcolm Christie got free on the left-hand corner of the Slovenian penalty box and sent a low cross towards goal, where Ameobi tapped home from a yard out into an empty net.
Playing against ten men, England were able to hold out for victory, and sub Ameobi will be particular pleased with his endeavours.
The Newcastle hit-man also hit a post, after good work from Konchesky, and the Charlton ace himself should have doubled the lead when wasting a good opportunity presented by Jermaine Pennant's pass.
