The Terriers needed a point to move out of reach of Carlos Carvalhal's side - and achieved that despite Marcos Alonso canceling out Laurent Depoitre's opener.
Huddersfield moved to 37 points with the draw at Stamford Bridge, taking them out of the reach of Swansea's 33.
It means that Swansea can now only catch Southampton on the final day of the season - but an inferior goal difference makes that scenario highly unlikely.
The Saints are three points clear of the Swans - with Mark Hughes' side sitting on 36 points.
It all means that if Swansea City are to stay in the Premier League for next season, they need to beat Stoke on the last day and hope that Southampton fail to move off that 36 points mark.
That would see Carvalhal's side move level with the Saints on points and then it would be down to goal difference.
The Swans have a goal difference of -27, a considerable way off Southampton's -18.
If they were to draw level on points with Southampton on the final weekend, they'd need to overcome a goal difference gap of 9 goals.
While the Saints are playing champions Manchester City, the odds are against the Swans staying up at Southampton's expense.
Though if the Swans beat Stoke 5-0 and Man City beat Southampton 5-0 the sun may come through the clouds.
It may be that the race is not always to the swift, nor the battle to the strong—but that’s the way to bet. Damon Runyon