The goal that started with a piece of brilliance from substitute Cesc Fabregas, who lobbed the ball over a napping defence for Andres Iniesta to feed substitute Navas, killed off Croatia's chances of a spot in the quarter-finals.
Spain topped the group on seven points with Italy, also advancing after a 2-0 win over Ireland, on five and Croatia with four. The Irish finished with zero.
It was, however, hardly a display that world champions Spain will want to remember. "The performance as a whole was not brilliant but we dominated," said coach Vicente del Bosque.
"It was a tough match and they managed to play their game as well," said Navas. "It was difficult to penetrate their defences."
Spain, initially sticking with forward Fernando Torres in the line-up after his two goals against Ireland, had the first chance with the 28-year-old's header in the opening minutes.
With the lion's share of possession in the first half, they had control of the game, keeping their opponents constantly on the backfoot but lacked the final decisive touch to score.
"We didn't have that many options going forward but that was also down to their virtues," said Del Bosque. "Our first touch let us down at times and that robbed us of speed and intensity."
Croatia grew more positive, especially after Italy scored in the other game with the result flashed across the big screens, and Rakitic should have scored but keeper Iker Casillas blocked his header and the rebound after Luka Modric's pinpoint cross.
But while Croatia demanded a penalty for the pull on Corluka, it was Spain who were rewarded when Fabregas's chip beat the offside trap and Iniesta found Navas for an easy goal.
"The game plan was to thwart Spain in the first half and have a go at them in the second," said Bilic. "It worked and we did everything right except score. We came here to go all the way and we failed because we lacked that tiny bit of luck that makes all the difference."
