Kenny Dalglish was left frustrated again after the Reds failed to make the most of their chances in the first half as Andy Carroll spurned the home side’s best opening in the fifth minute.
Swansea enjoyed a period of sustained pressure after the break, with the Reds fortunate to avoid conceding a goal, before Vorm prevented Liverpool from scoring a dramatic late winner.
The result keeps Liverpool in sixth place as Swansea move up to 10th in the table going into the international break.
Liverpool started brightly, looking to manufacture an early goal to settle their nerves.
And they could have led within five minutes. Charlie Adamâ€™s incisive pass cut open the Swansea defence, finding Stewart Downing.
The winger whipped a good cross into the area but Andy Carroll failed to convert, smashing his effort against the crossbar.
Dalglishâ€™s side continued to press but, as has been common throughout the Redsâ€™ slow start to the season at Anfield, they could not capitalise and the visitors began to create some chances of their own.
Pepe Reina was forced into his first save of the afternoon after 28 minutes when Wayne Routledge chipped a tempting ball into the area and the Spanish shot-stopper pushed Danny Graham’s effort round the post.
It prompted a reaction from the home side and SuÃ¡rez seized on a loose ball and unleashed a low shot which Vorm palmed wide.
With Liverpool pressing, Routledge was able to manufacture space for Joe Allen – and the midfielder swivelled past Martin Skrtel but curled his effort narrowly wide of Reinaâ€™s goal.
After the interval Dalglish substituted Jordan Henderson, who once again was underwhelming, for Dirk Kuyt in search of the crucial opening goal.
However, it was Swansea who came closest just after the break.
Nathan Dyer raced onto Routledgeâ€™s through-ball and looked certain to score – but Reina’s quick turn of pace saw him race from his line to deny the Swansea winger.
And Dyer was central to Swanseaâ€™s next chance on goal when his shot was spilled by Reina and Graham pounced, only to be denied by the reactions of Liverpoolâ€™s captain in goal.
Brendan Rodgersâ€™ men continued to press for a famous victory and Daniel Aggerâ€™s ill-timed challenge conceded a free-kick on the very edge of Reinaâ€™s area. Mark Gower stepped up but fired his free-kick harmlessly wide.
Gower then missed the visitors’ best chance of the second half. Another teasing run by Dyer left JosÃ© Enrique stranded and Dyer’s cross was headed back across goal by Graham.
But Gower, allowed to run into the box unmarked, wasted the chance as he sent his shot high into the Kop with the Liverpool defence in disarray.
Dalglish replaced Carroll with Craig Bellamy and the Welsh striker’s electric pace tormented the Swansea rearguard as gaps began to appear in the closing minutes of the game.
In the 88th minute, Liverpool thought they had netted a vital winner.
Glen Johnsonâ€™s cross was nodded down by Agger into the path of Kuyt, who scrambled the ball over the line – but the goal was chalked off as the striker was deemed to be offside.
There was still time for more drama. SuÃ¡rez produced a moment of magic to draw a fantastic save from Vorm and deep into stoppage-time the Swansea shot-stopper was on hand to deny Johnsonâ€™s improvised half-volley, athletically tipping the ball over the crossbar.
Liverpool: JosÃ© Reina, Martin Skrtel, JosÃ© Enrique, Glen Johnson, Daniel Agger, Charlie Adam, Jordan Henderson (Dirk Kuyt, 45), Stewart Downing, Lucas Leiva, Andy Carroll (Craig Bellamy, 75), Luis SuÃ¡rez
Swansea City: Michel Vorm, Angel Rangel, Garry Monk, Ashley Williams, Neil Taylor, Nathan Dyer, Leon Britton, Joe Allen, Mark Gower, Wayne Routledge (Scott Sinclair, 75), Danny Graham
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