Miami FC keeper Room pulled off a record-equalling 15 saves to keep out the South Americans in Group E.
The 37-year-old was in stunning form to match USA goalkeeper Tim Howard's record for the most saves in a World Cup game.
Howard made 15 saves against Belgium in 2014 in a match that went to extra time (also conceding twice in a 2-1 defeat) - but Room's clean sheet earned Dick Advocaat's side a historic draw.
His first save came after just three minutes and was arguably the best of the night as he kept out Enner Valencia from close range.
John Yeboah was next to be denied before Room saved again from Valencia among six first-half stops.
Room was even busier after the break as he denied Moises Caicedo, Pedro Vite, Kevin Rodriguez, Nilson Angulo, Gonzalo Plata and others.
Curacao did have a great opportunity of their own on the hour mark but Leandro Bacuna's strike was saved before Liviana Comenencia had an effort turned away.
Ecuador will rue their missed chances - with 27 attempts in total - and must beat group winners Germany in their final game to progress to the knockout stages, having lost to Ivory Coast in their opener.
Curacao - who lost 7-1 to Germany in their first game - could also qualify for the last 32 if they beat Ivory Coast in their final match.
Ecuador boss Sebastian Beccacece will be under scrutiny after their campaign began with just a point - and no goals - from their first two matches.
Their opening game was a hard-fought 1-0 loss to Ivory Coast, but Ecuador would have expected to beat Curacao, especially after the island nation were thrashed by Germany.
They can blame Room's stunning performance given they had an xG of 3.05, but it is now 39 attempts on goal across two games without scoring.
Under Beccacece Ecuador have been defensively resolute. They have conceded just eight goals in his 21 in charge which is helped by having top defenders like Arsenal's Piero Hincapie and Paris St-Germain's Willian Pacho.
But their defensive solidity seems to have come at a cost to their attack - they netted just 14 goals in 18 qualifying matches and are goalless so far at the World Cup.
Up next is Germany - and unless the four-time winners rest some key players given they have won the group, Ecuador face an uphill battle to reach the knockouts.