Kris Petersen proved the difference between the two sides with the only goal of the game midway through the second half.
The Edinburgh side had started the match brightly, with Callum Paterson and Danny Wilson both threatening with early chances but neither side could make a breakthrough before the interval.
Liverpool thought they had taken the lead after the restart when a Brad Smith header appeared to cross the line but the opener finally came on 64 minutes on the clock when Petersen slotted past Jamie MacDonald.
Controversy came when the referee demanded Ryan Stevenson be subbed after he reacted badly to a Jordan Rossiter challenge.
And there was worse to come for the Jambos as Callum Tapping was stretchered off with a knee injury late on.