Liverpool defended superbly throughout to record their fourth successive clean sheet and clinched victory with a 69th-minute goal by Raul Meireles.
In Sunday's other Premier League match, Birmingham City won 1-0 at fellow strugglers West Ham United.
Manchester United, beaten by bottom club Wolverhampton Wanderers on Saturday, lead the Premier League standings on 54 points. Arsenal, who blew a 4-0 lead to draw 4-4 in an incredible match at Newcastle United, are second on 50.
Manchester City are third on 49 with champions Chelsea and Tottenham Hotspur on 44. Liverpool moved to sixth on 38 and, a matter of weeks after flirting with the relegation zone, are now eyeing European qualification.
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