Osimhen nodded in straight after Iban Salvador’s opener for the underdogs.
Salvador’s classy first-half finish was his side’s only attempt on target and briefly threatened a significant upset in the second game of the tournament.
Jesus Owono made several second-half saves to deny the Super Eagles, and Osimhen missed after breaking clear.
The stalemate in the Group A match in Abidjan means both sides are two points behind Ivory Coast, who opened the tournament with a 2-0 win over Guinea-Bissau on Saturday.
Equatorial Guinea, who are 46 places below Nigeria in the world rankings, celebrated ecstatically after taking the lead, but conceded straight from the restart when Osimhen converted Ademola Lookman’s cross and spent much of the match under pressure.
Goalkeeper Owono was in inspired form but breathed a sigh of relief when African Footballer of the Year Osimhen misfired after being sent through by Fulham midfielder Alex Iwobi with 14 minutes remaining.
Zaidu Sanusi had lifted an early Super Eagles chance high over the crossbar when debutant Alhassan Yusuf’s cross found the full-back in space inside the penalty area.
And Osimhen, the leading marksman in qualifying for the tournament, also spurned a presentable opportunity with the game goalless, heading Ola Aina’s inviting delivery wide.
Three-time continental champions Nigeria are seeking to lift the trophy for the first time since 2013 but could not find a winner against opponents who had beaten holders Algeria on their way to the quarter-finals at the 2021 finals.
Owono tipped a low drive from Moses Simon onto his left-hand upright just before half-time and also kept out efforts from Sanusi, Osimhen and Aina as the second half wore on.
Equatorial Guinea return to the Alassane Ouattara Stadium when they face Guinea-Bissau on Thursday (14:00 GMT).
Nigeria take on Ivory Coast at the same venue in the subsequent game (17:00 GMT), pitting two teams considered strong contenders for the title against each other.