Emre Can could have found the opener for the Reds in the eighth minute with a free-kick from outside the box but his strike could not dip in time to find the target.
Virgil van Dijk had an opportunity to send the Reds into lead in the 22nd minute with a free-header inside the box but his effort sailed over the bar following a cross from Alex Oxlade-Chamberlain.
Mohamed Salah could have registered his name on the score-sheet in the 29th minute after being sent through on goal by van Dijk but the Egyptian skied his effort over the bar.
Jurgen Klopp's charges dominated the encounter and kept the Swans in their own half, but they lacked the killer-instinct upfront.
Alfie Mawson punished the Reds five minutes before halftime with a low right- footed strike inside the box after the visitors failed to clear their lines following a corner, 1-0.
On the hour mark, Salah came close to finding the reply from a dead-ball situation, but his curling strike was brilliantly tipped over the bar by Lukasz Fabianski.
Klopp made his first change 20 minutes from time by bringing on Adam Lallana in place of Alex Oxlade-Chamberlain.
Danny Ings almost got the equaliser with 13 minutes left to play after going clear but his low strike was smothered by Fabianksi.
The Reds kept the pressurising the Swans in the final minutes but they were unlucky in the dying minutes of stoppage time to see Roberto Firmino's header hit the upright.