A two-goal salvo from Chris Sutton helped Celtic book their place in the final of the Scottish Cup.
Sutton guides the ball past McKenzie
The Englishman helped himself to his first-ever goals in the competition on either side of the interval.
Henrik Larsson also got on the score-sheet, while Colin McMenamin grabbed a consolation for Livingston 10 minutes before time.
Larsson will now almost certainly play his final game for the Parkhead side in the final in May.
It was Livingston who created the first real opportunity after five minutes and would have gone ahead but for more heroics from Celtic's teenage goalkeeper David Marshall.
Derek Lilley exchanged passes with Fernando Pasquinelli but the keeper saved Celtic by blocking the Livingston man's shot from just eight yards.
At the other end, Roddy McKenzie saved Oscar Rubio's blushes by rushing off his line and gathering at the second attempt after his weak header had almost let Sutton in.
Celtic's Pearson (right) battles with McLaughlin
Larsson gave the Livi goalkeeper a test in the 17th minute, after Stephen Pearson dispossessed Pasquinelli and played the ball up to the striker, who turned and unleashed a right-footed shot which McKenzie dived to his left to save.
Livingston's defence was again carved open in the 26th minute but the Glasgow giants could not capitalise on a glorious chance.
Alan Thompson threaded the ball through to Sutton inside the box, but he seemed shocked to be in such a position and could only toe a shot straight at the alert McKenzie.
Pearson was also producing heroics at the other end and his slide inside the area prevented the ball from ending up at the feet of Lilley for an easy tap-in.
That proved to be the turning point as Celtic went ahead in the 37th minute through Sutton as McKenzie was horribly exposed.
Larsson held off the challenge of David McNamee and crossed to Stilian Petrov, who neatly cushioned his header into the path of Sutton to steer the ball into the net.
Moments later, Thompson limped off to be replaced by Manchester United-bound Liam Miller.
Pearson almost extended Celtic's advantage seconds before the break after he was superbly put through by Larsson but McKenzie again kept Livingston in the game.
Celtic stretched their advantage after the break after a disastrous defensive blunder by Livingston.
Thomspon (left) went off injured
Dorado and McNamee both went for the same ball and only succeeded in taking each other out and leaving Larsson with a free run at goal - and he coolly lobbed the ball over McKenzie.
Celtic put the result beyond doubt ten minutes later when Sutton grabbed his second.
Again there were bigger gaps in the Livingston defence than among the supporters.
Having been teed up by Larsson inside the box, he stopped, steadied himself and then rifled the ball past Andrews into the corner of the net with McKenzie nowhere.
There was concern for Martin O'Neill moments later as Sutton hobbled off to be replaced by Craig Beattie and he also took Larsson off for Johan Mjallby.
Livi substitute McMenamin gave the small travelling contingent something to smile about at last with a consolation in the 79th minute.
He had just lifted the ball over Marshall and wide a minute early but he made no mistake with a glancing header - in off the post - from seven yards.
Livingston: McKenzie, McNamee, Rubio, Andrews, Dorado, O'Brien, Makel, Lovell, McLaughlin, Pasquinelli, Lilley.
Subs: Main, McGovern, McMenamin, Snowdon, Brittain.
Celtic: Marshall, Varga, Balde, Valgaeren, Agathe, Lennon, Petrov, Pearson, Thompson, Larsson, Sutton.
Subs: Douglas, Lambert, Mjallby, Beattie, Miller.
Referee: K Clark (Scotland).