Essien, recently back from a long layoff with a knee injury, beat City keeper Shay Given in the 18th minute to seal the victory that left Chelsea four points behind leaders Manchester United who lost 4-1 at home to Liverpool on Saturday.
Champions United, who have played a game fewer than their rivals, have 65 points with Chelsea and Liverpool on 61.
Aston Villa's poor run continued in a 2-1 home defeat by Tottenham Hotspur which left them below fourth-placed Arsenal in the race for a Champions League spot.
Since Dutchman Hiddink replaced Luiz Felipe Scolari, Chelsea have gone seven matches unbeaten and after United's defeat on Saturday are still in the title race.
"We were happy with the win but if we had got the second goal we could have won three or four nil," Hiddink told the BBC.
"We were not anxious because defensively we were OK but nevertheless we should have killed off the game."
Hiddink said Drogba's injury was not serious but that Deco could be out for the season with a hamstring injury.
Big-spending City, who have won just once away in the league this season and are still not clear of relegation worries, were poor and could have been behind even before Ghana international Essien diverted in Frank Lampard's shot with his shin.
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