Lewandowski sent Barcelona ahead on the hour with a composed finish, but Osimhen levelled the scores 15 minutes later to leave the tie delicately poised ahead of the second leg in Spain on March 12.
The contest pitted the champions of Italy against the champions of Spain, but both have had huge problems this season, with the former currently languishing down in ninth position in the Serie A table, and Barcelona sitting third in the La Liga table, eight points off the leaders Real Madrid.
Napoli had their new manager on the touchline in the shape of Francesco Calzona, and he will have been pleased to claim a draw on the night considering the disappointment of the team's overall performance.
The first half-chance of the match came for Barcelona in the fourth minute, but 16-year-old Lamine Yamal's effort was always rising over the Napoli crossbar.
Yamal, who became the youngest-ever player to feature in the Champions League knockout stages, was again involved in the ninth minute, with the Catalan outfit in charge in the early exchanges, but the Spain international's left-footed strike brought a smart save from Napoli goalkeeper Alex Meret.
It was all Barcelona, and Lewandowski was next to try his luck in the 14th minute after being found by Pedri, but the experienced forward's effort was into the side-netting.
Meret again came to Napoli's rescue in the 22nd minute to deny Lewandowski, before keeping out an effort from Ilkay Gundogan, with the home side struggling to offer a threat of their own.
Lewandowski then glanced a Gundogan free kick wide of the target in the 33rd minute, but there was an improvement from Napoli in terms of possession in the closing stages of the first period.
The Italian champions did not manage a single attempt in the opening 45 minutes, which was an indication of their struggles, but it took them less than a minute to have an effort in the second period, with Matteo Politano heading wide of the target.
Meret made another smart save to deny Gundogan in the 49th minute, but Barcelona continued to carve out openings, and the breakthrough came on the hour through Lewandowski, who collected a clever pass from Pedri before finishing emphatically into the bottom corner.
Pedri tried his luck in the 66th minute, unleashing a rasping strike from the edge of the Napoli box, but Meret managed to make an uncomfortable save to prevent Barcelona from netting a quick-fire second.
Out of nowhere, Napoli managed to level the scores in the 75th minute, with Osimhen bursting into life, turning Inigo Martinez inside the Barcelona box before firing past goalkeeper Marc-Andre ter Stegen to spark jubilant scenes inside the famous stadium.
Suddenly it was Barcelona on the back foot, with Napoli sensing a second, which would have been harsh on the La Liga champions considering their overall performance on the night.
Andre-Frank Zambo Anguissa flashed a header wide of the post in the 85th minute, before Giovanni Simeone missed the target late on; the game's golden chance came in the final seconds, though, with Gundogan placing one just wide of the post from outside the box, as the match finished all square.