Two goals in a three-minute spell from David Murphy and Dean Shiels late in the first half set up the victory and Garry O'Connor added a third after half-time.
Hibs had the first chance after quarter-of-an-hour when stand-in skipper Guillaume Beuzelin was not picked up at a Derek Riordan corner, but he could not direct his header on target.
Riordan then beat the Livingston offside trap to connect with a Murphy cross but he could get no power in his volley and Roddy McKenzie saved easily.
Martin Scott had an even better chance at the other end shortly afterwards, but pulled his shot just wide with only Zibi Malkowski to beat.
But Livingston were grateful to Dave Mackay for a superb last-ditch touch when O'Connor looked set to score.
Hibs finished the first half strongly with two goals in a stunning three-minute spell.
Murphy cracked the first past McKenzie with a brilliant 20-yard shot and, before Livingston had time to recover, they were two down.
Scott Brown picked out Shiels and he slipped past two Livingston defenders before finding the net.
O'Connor almost scored a third before the break, but McKenzie slid out to block his shot.
The striker did not have long to wait. Eight minutes into the second half, he took a pass from Michael Stewart and drilled an angled shot past McKenzie to complete the scoring.