Celtic were given a home draw in the Scottish Cup for the first time since February 2009 this season, but a late goal from Steven Doris earned Arbroath a replay at Gayfield just when the Hoops appeared to have stuttered through thanks to an Alex Keddie own goal.
Adam Mathews settled the replay with a brilliant strike to take Celtic through on a 1-0 victory and a meeting with another lower division team in Raith Rovers.
The Hoops travelled to Starks Park and, despite a tricky surface, eased through with a 3-0 victory, thanks to a Kris Commons penalty and further goals from James Forrest and Charlie Mulgrew.
Celtic started March by avenging their League Cup defeat by St Mirren as goals from Joe Ledley and Anthony Stokes, either side of a strike from Esmael Goncalves, provided a 2-1 victory.
In a classic semi-final Dundee United turned the game around after conceding an early Commons goal, with strikes from Gary Mackay-Steven and Jon Daly.
Victor Wanyama levelled quickly and Commons put Celtic 3-2 ahead before Daly netted his second, but the tie was settled when fellow Irishman Anthony Stokes scored in extra-time as Celtic won 4-3.