The Nigerian headed in his fourth Blues goal from Willian's whipped free kick and Oscar made sure of the win with a swerving strike.
John Obi Mikel marked his 300th appearance for Chelsea with his fourth goal for the club to break the deadlock, rising to meet a Willian free kick in the 66th minute at Pride Park. Oscar then doubled Chelsea's lead four minutes later with a powerful strike from the edge of the penalty area that Derby goalkeeper Lee Grant was unable to keep out.
Ramires, who was booked for diving in the second half, also clipped the top of the crossbar late on as the seven-time FA Cup winners finished the third-round encounter strongly.
Chances were at a premium throughout with Oscar and Ramires coming closest for the visitors in the first half, while Derby, which contained its top-flight opponents well for the opening hour, rarely threatened Mark Schwarzer.
Victory for Jose Mourinho's side means Chelsea will face Stoke City at Stamford Bridge in the next round in the only all-Premier League tie.
The hosts were forced into two changes as injuries ruled out Craig Bryson (groin) and Johnny Russell (fractured cheekbone), so Jeff Hendrick and Simon Dawkins came in. Mourinho made six changes from the Chelsea side that beat Southampton on Wednesday as Schwarzer took over in goal from Petr Cech and Samuel Eto'o replaced Fernando Torres in attack.
Derby created the best chance of the opening exchanges with 13 minutes gone as it attacked down the left, but Jamie Ward was unable to meet Craig Forsyth's cross and Ashley Cole cleared the danger.
A mistake by Michael Keane in the 22nd minute saw the unmarked Eto'o released into the Derby half. Andre Wisdom and Jake Buxton muscled the Cameroon international off the ball, however, before he could get his shot away.
Chelsea finished the half the stronger, but wasted chances to break the deadlock through Ramires and Oscar. Clear-cut opportunities after the restart were few and far between before Chelsea took the lead just after the hour.
Mikel rose unmarked at the front post to meet a Willian free kick from the left and his flicked header beat Grant from five yards.
Oscar finally got his goal in the 70th minute with a well-struck effort from inside the penalty area. The Brazilian created room on the edge of the box and lashed a shot at goal that found a way past Grant.
Substitute Torres, who had replaced Eto'o, had two chances in the closing minutes of the match, but was denied by Grant on both occasions and the woodwork frustrated Ramires as Chelsea went in search of a third.