Ivory Coast raced into a 3-0 lead after 68 minutes but Israel fought back to 3-2 before Drogba struck from the spot seven minutes from time after being fouled to clinch the win.
Maor Melicksen added his second to make it 4-3 late on.
Israel were warming up for Euro 2012 qualifiers against Group F leaders Greece and Croatia next month but the encounter was a fairly tepid affair until the stroke of half-time.
The crowd were getting ready for refreshments when Ivory Coast opened the scoring through Yaya Toure who blasted a low shot into the net from 18 metres in the 45th minute.
Didier Ya Konan added a second goal for the Ivorians four minutes into first-half stoppage-time with a powerful strike from the right that left Israel goalkeeper Dudu Aouate helpless.
The powerful Ivory Coast side increased their lead in the 67th minute through substitute Moussa Kone who picked up the ball in the area and flicked it over the defence and Aouate.
Israel came to life in the last 15 minutes when striker Itay Shechter put them on the scoreboard after intercepting the ball in the Ivorian defence and breaking free to find the net.
Melickson further cut the arrears in the 79th after another defensive lapse but Drogba restored the African side's two-goal lead a minute later with his 48th goal in 73 internationals.
Melickson wrapped up the scoring in the 87th with a penalty but the defeat and poor defending leaves Israel coach Luis Fernandez looking for the right formula before they face Greece in Tel Aviv on September 2 and visit Croatia four days later.
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