ABIDJAN, Ivory Coast—Fighters aligned with Ivory Coast's President-elect Alassane Ouattara on Monday pushed into strongholds of Laurent Gbagbo, the incumbent who has resisted diplomatic efforts to dislodge him, signaling a violent new phase in the monthslong saga over a disputed election.

In the main city of Abidjan, witnesses said fighting erupted around the residence of Mr. Gbagbo's army chief of staff. The assault on the army chief's residence Sunday evening marks the first time an insurgency in Abidjan has spread beyond...