Chelsea duo inspire World Cup wins
Drogba grabbed the winning goal as the Ivory Coast ended Burkina Faso's unbeaten run in the African preliminaries with a 3-2 win in Ouagadougou.
His club mate Essien, meanwhile, proved pivotal in midfield as two goals from Dutch-based striker Matthew Amoah saw Ghana overpower Sudan 2-0 away in Khartoum.
Both Ghana and the Ivorians have a maximum nine points from three games and head their groups at the halfway point of the campaign to win a spot in South Africa for next year.
There was an important away win for Algeria in Zambia too while Nigeria got a potentially decisive point in Tunisia but Morocco's chances look bleak after they were held to a goalless draw at home by Togo.
Drogba's 70th minute finish ensured a second away win this month for the Ivorians, who have a three point lead in Group E.
Yaya Toure had put the away side into a 13th minute lead but Burkina Faso, who had been unbeaten in their eight previous qualifiers, came back through German-based midfielder Jonathan Pitroipa.
An own goal after half-time restored the Ivorian lead before Drogba added a cushion which they held onto despite Aristides Bance pulling a goal back.
Amoah struck early for Ghana in their Group D match and added a another midway through the second half to help Ghana also win away for the second time this month.
Algeria snatched a 2-0 win away at Zambia in Group C and are three points clear after goals from Majid Bougherra and Rafik Saifi.
Tunisia remain top in Group B but Nigeria are just two points behind them and have an easier run in the second half of the preliminary programme.
Nigeria weathered a first half storm from their hosts in Tunis but might have snuck an away win if Michael Eneramo had taken up a late chance.
Kenya beat Mozambique 2-1 in Nairobi in the same group for their first points of the final round group campaign.
Mounir El Hamdaoui missed a penalty as Morocco, who have qualified for four previous World Cup finals, gained only their second point in Group A in 0-0 tie with Togo in Rabat.
The top team from each of the five groups qualifies for the finals. Two more qualifying matches in Guinea and Mali are being played on Sunday.