Second-half goals from Joe Ledley, Georgios Samaras, Anthony Stokes and Scott Brown eased Celtic past 10-man Dundee United and into the semi-finals of the Scottish Cup.
A late own goal from Marius Zaliukas earned St Mirren a 2-2 draw and with Hearts in the quarter-finals of the Scottish Cup.
Marius Zaliukas' extra-time winner earned Hearts a 2-1 victory over St Johnstone in the fifth round of the William Hill Scottish Cup at McDiarmid Park.
Celtic have been handed yet another away tie in the quarter-finals of the Scottish Cup.
Rangers plunged deeper into crisis after crashing out of the Scottish Cup as goals from Gavin Gunning and Johnny Russell secured a 2-0 win for Dundee United at Ibrox.
Scott Brown found the target for the third game in a row as Celtic overcame a battling Inverness Caledonian Thistle 2-0 to reach the quarter finals of the Scottish Cup, while Motherwell crushed Morton and Hibernian edged out Kilmarnock.
Celtic could face another meeting with cup bogey side Inverness in the fifth round of the William Hill Scottish Cup.
Holders Celtic will face Peterhead in the Scottish Cup fourth round in January following today's draw at Hampden Park.
Celtic manager Neil Lennon is relishing a return to the Hampden technical area following the end of his touchline ban.
Celtic could face a repeat of their Scottish League Cup semi-final against Aberdeen following the draw for the last four of the Scottish Cup.