Zinedine Zidane's side conceded an early goal at the Santiago Bernabeu but responded well to secure a 3-1 victory in the Round of 16 first leg.
Insigne latched onto a through ball from Marek Hamsik and with extraordinary awareness, saw Navas out of position and produced a first-time shot which curled and bounced around the Costa Rican.
10 minutes later, however, and Real had equalised when Dani Carvajal's sent in a splendid floating cross with the outside of his right foot which allowed Karim Benzema to rise above his marker and head home emphatically.
Early in the second half and the home side had edged ahead in fine style when Toni Kroos hit a lovely first-time shot to the left of Pepe Reina from the edge of the area, following a brilliant run and pass from Cristiano Ronaldo.
And Real settled the tie sublimely on 54 minutes when Casemiro hit a stunning volley from range which seemed to surprise even the scorer himself.
While it was not the start Real would have expected, they rallied to come away with a fairly convincing win ahead of the return bout in Naples on March 7.
From a Napoli point of view, Maurizio Sarri will not be too dejected as his side managed to score a crucial away goal to take back with them to the Stadio San Paolo.
2016/17 UEFA Champions League - Round of 16 first leg