A tale of two penalties ended with the Netherlands claiming three precious points in Athens thanks to a late penalty by Virgil van Dijk.
Wout Weghorst missed from the spot in the first-half with a tame shot that epitomised the toothless Dutch attack, with midfielder Tijjani Reijnders posing the biggest threat.
But with the match seemingly destined to end in stalemate, the captain made no mistake when Oranje were awarded a second penalty in the 90th minute.
Ronald Koeman’s side can secure the second qualifying spot by winning either of their final group games against the Republic of Ireland in Amsterdam and away to Gibraltar, while Greece’s only remaining game is at home against group winners France.
The coach made two changes to the team that lost to France on Friday, bringing in Steven Bergwijn for Joey Veerman and opting for Mats Wieffer, who came on as a second-half substitute in Amsterdam, instead of the man he replaced, Marten de Roon.
Weghorst passed a fitness test to spearhead a three-pronged attack with Steven Bergwijn and Xavi Simons, while Reijnders and Quilindschy Hartman retained their places after impressing on their debuts against the French.
A relieved Van Dijk said Greece had been tough to break down: “We created chances with the [first] penalty and a couple of times on the counter.
“At the end they threw on two extra strikers and gave us problems, but the penalty went in and we got over the line.”
After a cagey start the Dutch gradually took control and had a strong case for a penalty after 19 minutes when Giorgos Masouras looked to have touched the ball with his hand as he steered it behind the goal-line.
From the resulting corner Reijnders engineered the first clear chance when he fired wide from an unmarked position outside the area. Minutes later the midfielder tried again from an almost identical position from the next corner, this time drawing a save from Odysseas Vlachodimos.
Four minutes later Oranje had their first a penalty as Konstantinos Koulierakis hauled down Van Dijk, but Weghorst struck his spot-kick too close to Vlachodimos, who gathered easily.
Reijnders had another clear chance from range before Steven Bergwijn collected an outstanding long pass from Van Dijk, but his first-time shot was turned behind by the Greek goalkeeper.
Vlachodimos had to save smartly again from Bergwijn just after half-time when the winger combined well with Weghorst on the counter, hitting a low shot that the goalkeeper palmed wide.
Greece had rarely threatened up to that point, but could have snatched the lead when Nathan Aké was left stranded by Fotis Ionnadis, only for the striker to shoot into the side netting.
At the other end Reijnders wriggled his way into the area and hit a shot that Vlachodimos could only block, before Simons squirted the rebound wide.
Weghorst made way for another debutant, Ajax’s Brian Brobbey, while Veerman came on for Simons as Koeman sought a more direct approach.
Bart Verbruggen, who had barely stretched his legs in the Dutch goal, had to be alert 10 minutes from time when a shot by Petros Mantalos was deflected into the path of substitute Giorgos Giakoumakis.
The Greeks looked sharper in the closing stages as they brought on former VVV Venlo striker Giorgos Giakoumakis and AZ Alkmaar’s top scorer Vangelis Pavlidis.
Ioannidis put a header wide after substitute Micky van de Ven gave the ball away on the right-hand side, before Verbruggen easily gathered a long-range shot by Taxiarchis Fountas.
But just as it looked as if Oranje would have to settle for a point, they were given a lifeline when Vangelis Pavlidis pulled down Denzel Dumfries on the edge of the area.
Referee Alejandro Hernández was called over to check if Brobbey had checked a Greek defender just beforehand, but the penalty stood and Van Dijk dispatched it straight down the middle to salvage a vital win.
Ireland beat Gibraltar 4-0 in Faro in the other Group B game, with both sides already eliminated.
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