Double Dutch late show floors Mexico
Netherlands' forward Klaas-Jan Huntelaar celebrates after scoring a penalty during a Round of 16 football match between Netherlands and Mexico at Castelao Stadium in Fortaleza during the 2014 FIFA World Cup on June 29, 2014 - by Damien Meyer
In a match conditioned by the baking heat and humidity in Fortaleza, Giovani dos Santos looked to have Mexico on the way to their first quarter-final appearance on foreign soil when he opened the scoring three minutes into the second-half.
However, Sneijder levelled with a thumping volley from the edge of the area two minutes from time before Huntelaar converted from the spot in stoppage time after veteran and Mexico captain Rafael Marquez had upended Arjen Robben.
Louis van Gaal's men will now face Costa Rica or Greece in the last eight in Salvador on Saturday.
"The players showed they had faith and belief until the very end," said Van Gaal.
"The humidity was not in our favour so when you see that until the very last minute we were fresher and fitter than the Mexicans that is a big compliment to my players.
"Not only did they have belief but physically they prepared so well to play this match. That of course gives us enormous confidence going forward."
The Netherlands had been disrupted by an early injury to Nigel de Jong as he hobbled off to be replaced by Bruno Martins Indi.
Mexico's Hector Herrera then had the clearest opening of the first 45 minutes as he collected Oribe Peralta's pass on the edge of the area and cut inside before dragging a right-foot shot inches wide.
Both sides were then handed the chance to cool off for three minutes at the half-hour mark for the first time at a World Cup due to the searing temperatures.
Cillessen then made a fine stop to prevent dos Santos opening the scoring with a low effort.
However, the Netherlands were left aggrieved just a minute before the break when Robben appeared to have been clipped by Hector Moreno inside the area.
Mexico got off to a flying start in the second-half as Dos Santos held off a challenge from Daley Blind to thump home his first international goal in a competitive match for two years from 25 yards.
Peralta nearly made it 2-0 eight minutes later with a curling effort that Cillessen did well to claim at the first attempt.
Moments later it was Guillermo Ochoa who made an incredible stop to prevent Stefan de Vrij levelling as he turned the defender's volley onto the post from point-blank range.
Sneijder then saw a deflected effort fly inches past as the Netherlands went in search of an equaliser.
Ochoa had to come to his side's rescue again 17 minutes from time when Robben escaped the clutches of Marquez but saw his low effort turned behind by the Mexican stopper.
Dutch boss Louis van Gaal then surprisingly replaced captain Van Persie with Klaas Jan Huntelaar.
However, his decision was to prove inspired as firstly Huntelaar turned Robben's corner in the direction of the lurking Sneijder to power home from the edge of the box on.
And after Robben was adjudged by Portuguese referee Pedro Proenca to have been clipped by Marquez in the fourth minute of stoppage time, the Schalke striker coolly stroked home the resulting penalty to eliminate Mexico at the last 16 stage for a sixth consecutive World Cup.