The striker struck twice in the second half as the visitors deservedly grabbed a 2-2 draw in Group C at the King Power Stadium.
Ayoze Perez's first goal since March and Harvey Barnes' second-half goal gave the Foxes a commanding lead.
But Osimhen grabbed a classy lifeline for the visitors and then headed in a late leveller, with Leicester's frustrations compounded when Wilfred Ndidi was sent off in injury time.
Police and stewards had to control a disturbance between the home and travelling fans at full time with missiles thrown between supporters.
Napoli controlled much of the game and were almost left to rue several missed chances with Osimhen, Hirving Lozano, Piotr Zielinski and Kevin Malcuit all wasteful in the first half.
The Foxes - who only introduced Jamie Vardy as a late substitute - go to Legia Warsaw, who won at Spartak Moscow on Wednesday, in two weeks.
Leicester and Napoli started the tournament as joint favourites to lift the trophy in Seville in May and went at each other as if to prove their credentials.
Napoli threatened first and Kasper Schmeichel spilled Osimhen's early drive from distance before David Ospina came to the visitors' rescue.
The former Arsenal goalkeeper flung himself at Barnes' feet to turn his close-range shot wide after Perez bullied his way through and Kelechi Iheanacho helped on his cross.
But Ospina could not thwart the Foxes for long and they grabbed the opener after just nine minutes.
It owed much to Barnes' direct running when the winger collected the ball on the left from Ryan Bertrand's clearance.
He burst forward, swapped passes with Patson Daka and delivered a deep cross for Perez to volley in at the far post.
Napoli responded well and the lively Osimhen slashed wide before teeing up Malcuit to shoot over after 29 minutes.
Osimhen continued to be a menace and, when he beat Jonny Evans to cross, only a combination of Timothy Castagne and Schmeichel kept out Zielinski.
Napoli, who had won all three Serie A games this season, had slowly taken control and skipper Lorenzo Insigne drilled wide as the Foxes struggled to regain a rhythm.
The Italians were in command but were becoming wasteful with Nigeria international Osimhen firing another opportunity over before Schmeichel ensured Leicester went into the break ahead.
The Foxes had lived dangerously and survived another warning when Malcuit's cross found Lozano drifting into space and his header was parried by Schmeichel.
Leicester, playing their first home European game with fans since their Champions League quarter final with Atletico Madrid in 2017, needed a remedy.
Boss Brendan Rodgers recognised it, replacing Perez and Jonny Evans with Youri Tielemans and Caglar Soyuncu at the break, but Leicester initially failed to stem the tide.
Yet, the Foxes thought they had doubled their lead after 59 minutes.
Tielemans was involved, poking the ball through to Daka for the striker to drill in, only to be denied his first Leicester goal by a marginal VAR offside call.
Five minutes later, though, the Foxes did find their second goal.
Iheanacho won the ball in midfield and his perfect pass found Barnes on the left for the winger to tease Malcuit and drill into the corner.
Napoli had paid the price for their profligacy but they pulled a goal back in style with 21 minutes left.
A slick move on the edge of the box involving Insigne, Elif Elmas and Fabian Ruiz ended with Osimhen holding off Jannik Vestergaard and lifting the ball over Schmeichel from six yards.
Unsurprisingly, Napoli pressed for the leveller and Schmeichel turned Elmas' shot away but the Foxes could not hang on.
Osimhen had been a constant threat and he netted again with three minutes left to level, heading in Matteo Politano's cross from eight yards.
Ndidi was then dismissed in stoppage time for collecting a second yellow card after he pulled back Adam Ounas.