Chelsea, desperate for all three points after the champions were held to a 1-1 draw by Arsenal on Sunday, needed Drogba to come off the bench and poke the ball home in the 74th minute.
Arsenal moved up to third place after their Togo striker Emmanuel Adebayor scored in the 88th minute to give them a 1-0 victory at Wigan Athletic in the night's other game.
Manchester United are top with 44 points, Chelsea have 39 and Arsenal are third on 29 after 17 games.
Chelsea's performance at Stamford Bridge was at best workmanlike and at times poor against an injury-ravaged Newcastle side.
Chelsea coach Jose Mourinho said: "They didn't play very well, but at halftime I think they understood the message -- and the message was 'we cannot lose points today'.
"Sometimes you go with quality, sometimes you go with your heart -- and I think everyone went with their heart," he told Sky Sports News.
Mourinho played down the notion that Chelsea had sent United a message, a few weeks after coming back for a 1-1 away draw with Alex Ferguson's men.
"They (United) are not silly, they know what football is about, the manager and some of the players have a lot of experience. They know how strong we are, because they felt our second half at Old Trafford in their skin.
"And they know we are ready for the challenge."
There were rare starts for striker Salomon Kalou and Shaun Wright-Phillips but both fluffed first-half chances and the England winger was replaced at halftime.
Newcastle's French midfielder Antoine Sibierski hit the bar with an early header from a James Milner corner, with Chelsea's third-choice keeper Hilario making a poor start.
DROGBA FACTOR
But Drogba came on for the restart and the Ivory Coast striker soon added bite to what had been a pedestrian attack, further strengthened by Andriy Shevchenko.
The breakthrough came when Arjen Robben provided the pass, Shevchenko skewed a shot and the ball landed at the feet of Drogba, who prodded home his 15th goal of the season in all competitions.
Arsenal struggled at Wigan but finally came to life after a dour first half in which the hosts enjoyed most of the possession.
Arsenal manager Arsene Wenger said: "It was a difficult game. It could have gone for them but in the end it went for us. We needed to be really strong to come away with the three points."
Chances fell to both sides after the break before Arsenal's Spanish substitute Cesc Fabregas provided a great ball behind the Wigan defence for Adebayor to beat the offside trip and then keeper Chris Kirkland.