The Manchester City striker opened the scoring midway the first half, while the Arsenal midfielder, who created the first goal, made sure of the victory on 62 minutes.
It was the first Ecuador match since the death of Christian Benitez, who was capped 58 times for his country, and both teams paid tribute to the former Birmingham City forward by stopping the game for 60 seconds after 11 minutes.
It did not take long for the visitors to make their mark on their game. David Silva struck the outside of the far post with a scuffed left-footed effort with barely three minutes gone, while Negredo sent a volley over the bar five minutes later, following a darting run and cross by Cristian Tello.
Spain hit the woodwork again midway through the first half, as Andres Iniesta drove a low right-footed shot against the inside of the post.
The hosts had a chance of their own to break the deadlock when Felipe Caicedo appeared unmarked at the far post to meet a free-kick, but the striker headed well wide of goal.
It was left to Spain to open the scoring a minute later, and although it was Negredo who provided the finishing touch, the goal came about due to the excellent work of Cazorla.
The 28-year-old drove through midfield and played a one-two with Silva before taking the ball on his chest and volleying at goal. Ecuador stopper Maximo Banguera parried the shot, but Cazorla followed up his effort to provide a low cross to Negredo, who flicked the ball high into the net with his heel to score his seventh goal for his country.
Spain captain Iker Casillas was called into action just before the break, diving to his left to tip a powerful drive from Segundo Castillo over the bar.
Ecuador came closest to equalising on the hour mark, Enner Valencia heading a cross from Cristian Noboa against the far post, to the relief of a stranded Victor Valdes, who had replaced Casillas.
As in the first half, Ecuador’s best chance preceded a Spain goal, but on this occasion it was Cazorla who finished the move, running onto a backheel by debutant Tello to strike a low shot into the far corner.
Atletico Madrid midfielder Koke and Real Sociedad defender Inigo Martinez also made their debuts for Spain when they came on in the second half.
Substitute Fernando Llorente had the final chance of the game, but was unable to get himself on the score-sheet, steering a header several feet wide of the near post.