The Ivory Coast forward finished off a flowing counterattack in the 75th minute with a shot that squirmed into the far corner, earning Chelsea a replay at Stamford Bridge.
Everton was in control for long periods at Goodison Park but only had one goal to show for its dominance, with France striker Louis Saha heading home powerfully from a corner by Leighton Baines in the 62nd for a seventh goal in his last eight games against the London club.
Chelsea came close to snatching a win when Brazil midfielder Ramires hit the post in the 86th, while substitute Everton striker Jermaine Beckford forced an acrobatic save out of Petr Cech in the final minute.
Jack Rodwell, Marouane Fellaini and Seamus Coleman had all earlier gone close for Everton, which lost 2-1 to Chelsea in the 2009 final at Wembley.
Chelsea has won three of the last four FA Cups, including the last two.
In the first of two other all-Premier League matches on Saturday, Aston Villa defeated Blackburn 3-1 at Villa Park.
Ciaran Clark, veteran former France winger Robert Pires and Nathan Delfouneso scored Villa’s goals, all of them coming in the first half. Nikola Kalinic had equalized for Blackburn in the 18th, while Villa had to survive the last 28 minutes with 10 men following the sending-off of Nathan Baker for a dangerous tackle on David Hoilett.
Second-tier League Chmapionship side Swansea was upset 2-1 by Leyton Orient, which plays in English football’s third division.
Unbeaten Premier League leader Manchester United visits third-tier side Southampton later on Saturday.