Willian opened the scoring with a well-placed free-kick after five minutes and the Brazilian added his second from the spot two minutes before the break.
Harry Kane had headed in from Christian Eriksen's cross to equalise for Spurs, who pulled level again from Dele Alli.
Substitute Eden Hazard put Chelsea in front, before Matic fired in the fourth to keep them on course for the double.
Chelsea will face the winners of Sunday's semi-final between Manchester City and Arsenal in the final on Saturday, 27 May which will be their 12th FA Cup.
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