After two dour draws in their opening matches, England progressed in second place in Group C behind the United States, who came top by virtue of goals scored, after a much-improved, at times panicky, performance in Port Elizabeth.
Midfielder Frank Lampard said history had shown that results of group matches were insignificant in the knockout rounds and England were still contenders to lift the World Cup.
They next face Group D winners Germany in Bloemfontein on June 27.
"Our performance has given us a lot to build on. We've seen many times with big teams that you often start these tournaments a bit cagey but come good and strong," he said.
"It starts here - it's a knockout. How you played in the group didn't mean anything ... it's pure knockout football.
"It's about performing. When you're only four games away from something special, it can focus the mind," Lampard added.
Striker Jermain Defoe was recalled to the starting lineup and justified his selection after 23 minutes when he converted a delightful cross from James Milner, who was also drafted in to reenergise the flagging side.
IMPERFECT PERFORMANCE
England were always in control against the Slovenians, with Wayne Rooney, Steven Gerrard and John Terry standing out in an action-packed first half.
The performance was far from perfect, however, with plenty of missed chances, some wayward passing and a few panicky moments in defence. Rooney showed promise but was still far from the scintillating form he showed last season in the Premier League.
Lampard said England's players had accepted the tide of criticism about their poor start but had risen to the occasion and showed mettle when it counted most.
"When you get to this stage, you have to be big enough to be stand up to it, you can't have off days," he said.
"From the start there was a determination, a big feeling. There was something about us. If you take a bit of stick you have to react and there was determination in the way we played."
Joe Cole, who was widely tipped to start but came on as a second-half substitute, said England could regain the form that steered them through qualifying with nine wins from 10 games.
"Coming into a game with so much tension is tough," he added. "We'll play with more confidence because we rediscovered the spirit. It's a nice start but now we need to push on and try to win this World Cup."
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