The Brazilian midfielder thundered a spectacular right-foot shot into the net early in the second half to complete Real's comeback, adding to efforts either side of halftime from Karim Benzema and Toni Kroos on Wednesday.
Holders Real, who are chasing a 12th European title and eyeing a seventh straight quarter-final appearance, will travel to Naples for the return leg on March 7.
"We'll go there to a difficult ground where we'll suffer and it's still an open game," Real coach Zinedine Zidane told reporters.
"We need to go there and finish it off with our 3-1 advantage."
Real were defending an enviable home record in the Champions League, having gone 11 games undefeated at the Bernabeu, but were facing a Napoli side in form, unbeaten in their last 18 matches in all competitions.
At that point it looked like it could be a long evening for Napoli, but the Italians were in no mood to lie down and took the lead to cap a frenetic opening after eight minutes.
Insigne was fed through by Marek Hamsik and spotted Real keeper Navas out of position. Quick as a flash and with two defenders closing in, he bent a shot into the corner from 35 metres.
The lead lasted 10 minutes as Real drew level when Dani Carvajal's arrowed cross was headed powerfully into the net by Benzema, his 51st Champions League goal, lifting him into fifth on the all-time scorers list.
The remainder of the half was spent mainly in Napoli territory as Real continued to press, with Cristiano Ronaldo blasting over after being set free by Luka Modric and Benzema clipping the outside of the post when through on goal.
Four minutes later Casemiro took the game, and perhaps the tie, away from Napoli with a stunning right-foot volley from 25 metres that left Reina flailing.
"We faced the best team in the world, but it's a pity that we made so many mistakes with the ball because we usually don't do that," said Napoli coach Maurizio Sarri.
"The return leg? We will try to win 2-0, but I don't think Madrid would agree with that."