The game, which was due to be played in front of an empty stadium due to new restrictions enforced from Boxing Day, is yet another to be called off after the Hatters' visit to Reading was last weekend.
A statement on the Town official website said: "Swansea informed the EFL that taking into account the injuries and positive COVID-19 cases within their squad, the club does not have a sufficient number of players available in order to fulfil the fixture.
"In light of the increased rate of transmissions from the Omicron variant of COVID-19, the EFL will review the circumstances surrounding the postponement and work with affected clubs.
"A rearranged date for the fixture will be confirmed in due course."
Swansea themselves released a statement saying: "Swansea City’s Championship fixture against Luton Town has been postponed.
"This is due to an additional number of positive COVID-19 cases within the playing squad.
"The club has worked hard to try and ensure it would be able to fulfil the fixture, however, taking into account injuries and positive Covid-19 cases, the club does not have a sufficient number of players available.
"Any rearranged date will be confirmed in due course, while Swansea City will automatically refund supporters who have purchased streaming passes for this fixture."
Speaking about the fact there would have been no supporters present in his press conference this afternoon, Luton boss Nathan Jones had said: "The First Minister (Mark Drakeford) will obviously know far more than what I do but it is disappointing that we can’t have Luton fans there first and foremost.
"With the greatest of respect I’m not concerned about Swansea fans but I can’t criticise God’s country for shutting down because they must have good reason.
"This thing isn’t going away and what has to be considered is how dangerous it is, how potent it is, what damage it is doing before shutting down and taking away years from people."