Everton's opening goal was supplied by Cenk Tosun after the striker featured in a Blues' starting line-up for the first time since November 2019.
Rotherham levelled when Matt Olosunde found the corner of Robin Olsen's net with an accurate finish 11 minutes after half-time.
“The FA Cup is very important, it is a priority for us," said Ferguson, a member of the side which won Everton's fifth FA Cup in 1995.
“It is a chance win a cup for the fans so it is a big competition for us.
“We will keep putting out strong teams and take it really seriously – and we'll go as deep as we can.
“It was a tough match, I thought we started the game well… we had the best of the first 25 minutes but Rotherham came back into the game and pushed us all the way.
“They got a corner in the last minute [of normal time] and you fear the worst.
“There was a bit of relief at the end because it was a hard match.
“There are positives: we're in the next round, which is the main thing, and we are pleased to see Lucas back and get minutes under his belt.
“Hopefully he can recover for the game on Tuesday."
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