1959-60 Scottish Cup
| Final | ||||||
| Rangers | 2 | 0 | Kilmarnock | Sat 23 Apr 1960 | Details | |
| Semi Final | ||||||
| Rangers | 1 | 1 | Celtic | Sat 02 Apr 1960 | Details | |
| Kilmarnock | 2 | 0 | Clyde | Sat 02 Apr 1960 | Details | |
| Rangers | 4 | 1 | Celtic | Wed 06 Apr 1960 | 1st Replay | Details |
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| 4th Round | ||||||
| Rangers | 3 | 2 | Hibernian | Sat 12 Mar 1960 | Details | |
| Eyemouth United | 1 | 2 | Kilmarnock | Sat 12 Mar 1960 | Details | |
| Ayr United | 0 | 2 | Clyde | Sat 12 Mar 1960 | Details | |
| Celtic | 2 | 0 | Partick Thistle | Sat 12 Mar 1960 | Details | |
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| 3rd Round | ||||||
| Ayr United | 4 | 2 | Airdrieonians | Sat 27 Feb 1960 | Details | |
| Partick Thistle | 3 | 2 | Queen Of The South | Sat 27 Feb 1960 | Details | |
| Kilmarnock | 2 | 0 | Motherwell | Sat 27 Feb 1960 | Details | |
| Eyemouth United | 3 | 0 | Cowdenbeath | Sat 27 Feb 1960 | Details | |
| Stenhousemuir | 0 | 3 | Rangers | Sat 27 Feb 1960 | Details | |
| East Stirlingshire | 0 | 3 | Hibernian | Sat 05 Mar 1960 | Details | |
| Clyde | 6 | 0 | Queen's Park | Sat 05 Mar 1960 | Details | |
| Elgin City | 1 | 2 | Celtic | Sat 05 Mar 1960 | Details | |
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| 2nd Round | ||||||
| Elgin City | 5 | 1 | Forfar Athletic | Sat 13 Feb 1960 | Details | |
| Aberdeen | 0 | 2 | Clyde | Sat 13 Feb 1960 | Details | |
| Alloa Athletic | 1 | 5 | Airdrieonians | Sat 13 Feb 1960 | Details | |
| East Stirlingshire | 2 | 2 | Caledonian | Sat 13 Feb 1960 | Details | |
| Cowdenbeath | 1 | 0 | Falkirk | Sat 13 Feb 1960 | Details | |
| Dunfermline Athletic | 2 | 3 | Stenhousemuir | Sat 13 Feb 1960 | Details | |
| Montrose | 2 | 2 | Queen's Park | Sat 13 Feb 1960 | Details | |
| Dundee United | 2 | 2 | Partick Thistle | Sat 13 Feb 1960 | Details | |
| Motherwell | 6 | 0 | Keith | Sat 13 Feb 1960 | Details | |
| St Mirren | 1 | 1 | Celtic | Sat 13 Feb 1960 | Details | |
| Stirling Albion | 3 | 3 | Queen Of The South | Sat 13 Feb 1960 | Details | |
| Eyemouth United | 1 | 0 | Albion Rovers | Sat 13 Feb 1960 | Details | |
| Rangers | 2 | 0 | Arbroath | Sat 13 Feb 1960 | Details | |
| Queen Of The South | 5 | 1 | Stirling Albion | Wed 17 Feb 1960 | 1st Replay | Details |
| Partick Thistle | 4 | 1 | Dundee United | Wed 17 Feb 1960 | 1st Replay | Details |
| Peebles Rovers | 1 | 6 | Ayr United | Mon 22 Feb 1960 | Details | |
| Hearts | 1 | 1 | Kilmarnock | Mon 22 Feb 1960 | Details | |
| Kilmarnock | 2 | 1 | Hearts | Wed 24 Feb 1960 | 1st Replay | Details |
| Celtic | 4 | 4 | St Mirren | Wed 24 Feb 1960 | 1st Replay | Details |
| Caledonian | 1 | 4 | East Stirlingshire | Sat 27 Feb 1960 | 1st Replay | Details |
| Celtic | 5 | 2 | St Mirren | Mon 29 Feb 1960 | 2nd Replay | Details |
| Hibernian | 3 | 0 | Dundee | Mon 29 Feb 1960 | Details | |
| Queen's Park | 1 | 1 | Montrose | Mon 29 Feb 1960 | 1st Replay | Details |
| Queen's Park | 2 | 1 | Montrose | Tue 01 Mar 1960 | 2nd Replay | Details |
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| 1st Round | ||||||
| Berwick Rangers | 1 | 3 | Rangers | Sat 30 Jan 1960 | Details | |
| Aberdeen | 0 | 0 | Brechin City | Sat 30 Jan 1960 | Details | |
| Stenhousemuir | 2 | 0 | Rothes | Sat 30 Jan 1960 | Details | |
| Keith | 3 | 0 | Hamilton | Sat 30 Jan 1960 | Details | |
| Kilmarnock | 5 | 0 | Stranraer | Sat 30 Jan 1960 | Details | |
| East Fife | 0 | 2 | Partick Thistle | Sat 30 Jan 1960 | Details | |
| Queen Of The South | 2 | 1 | Dumbarton | Sat 30 Jan 1960 | Details | |
| Queen's Park | 2 | 0 | Raith Rovers | Sat 30 Jan 1960 | Details | |
| Clyde | 2 | 0 | Third Lanark | Sat 30 Jan 1960 | Details | |
| Dunfermline Athletic | 1 | 1 | St Johnstone | Sat 30 Jan 1960 | Details | |
| St Mirren | 15 | 0 | Glasgow University | Sat 30 Jan 1960 | Details | |
| Greenock Morton | 0 | 1 | East Stirlingshire | Sat 30 Jan 1960 | Details | |
| Albion Rovers | 2 | 1 | Tarff Rovers | Sat 30 Jan 1960 | Details | |
| Brechin City | 3 | 6 | Aberdeen | Wed 03 Feb 1960 | 1st Replay | Details |
| St Johnstone | 1 | 4 | Dunfermline Athletic | Wed 03 Feb 1960 | 1st Replay | Details |
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