The Toffees have a dreadful FA Cup record in recent seasons and they can count themselves lucky to avoid a giant killing in this one.
Forgotten man Cenk Tosun got the ball rolling for Ancelotti inside the first ten minutes as he latched onto Anthony Gordon’s reverse pass to clip home.
However, the Premier League side failed to build on that before the break as Rotherham pair Matthew Olosunde and Daniel Barlaser tested Robin Olsen.
And with Everton unbale to carve out a second goal after the restart the visitors took full advantage as Olosunde slid home following Tom Davies error.
Tosun did have the ball in the net before the end of normal time but the Turkish international was left frustrated by a VAR offside call.
Ancelotti likely gave his players a rollocking ahead extra minutes as his side slowly wrestled back control before Abdoulaye Doucoure sealed the win.
Full Time from Goodison Park it’s Everton 2 Rotherham United 1