Chelsea, who brought on Drogba, Frank Lampard and Michael Ballack in the second half, broke the deadlock on 79 minutes when the Ivory Coast striker scored from the edge of the area after Arjen Robben had been fouled.
"Sometimes you have a good period like this," said Drogba, who has notched three goals in three games.
"It is very difficult to be on the bench and come on. The pace is very high," he told Sky Sports.
Newcastle striker Obafemi Martins had come within a whisker of opening the scoring on 27 minutes when his superb 25-metre shot hit the underside of the bar and bounced off the line.
Andriy Shevchenko came close for Chelsea on the stroke of halftime when his angled drive clipped the far post while Lampard also struck the woodwork with a chip in stoppage time.
Tottenham were given a scare by second division bottom club Southend, who beat Manchester United 1-0 in the previous round.
The first 90 minutes at White Hart Lane ended goalless and it was another 25 minutes before England striker Jermain Defoe scored the winner from a Mido pass.
Fourth division Wycombe Wanderers shocked Premier League strugglers Charlton Athletic 1-0 in the first quarter-final on Tuesday.
Liverpool's tie with Arsenal was postponed because of fog and has been rearranged for January 9.