Celtic took the lead in the first minute when Gary Mackay-Steven prodded Griffith's pass home, but Ross County pulled level in the 14th minute when Ambrose fouled Alex Schalk in the box and was sent off. Martin Woods scored from the spot.
Paul Quinn headed County ahead on his second debut for the club and Schalk made it 3-1 before Griffiths saw his penalty saved by Scott Fox.
In the final, Ross County will face Hibernian, who beat St Johnstone 2-1 in the other semi on Saturday, on March 13.
Celtic are six points clear of Aberdeen in the Scottish Premiership title race and face non-League East Kilbride in the last 16 of the Scottish Cup next week.
(Reporting by Mike Collett; Editing by Ken Ferris)
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