In the final moments of the game, as Prescott tried to stop the clock by throwing the ball to the turf, umpire Ramon George collided with the quarterback and time expired before another play could be run.
After the game, videos on social media showed fans throwing debris aimed at the officials as they walked off the AT&T Stadium field in Arlington, Texas -- some almost hitting defensive end DeMarcus Lawrence.
When Prescott was asked about the fans' actions, he initially expressed his disappointment thinking that they were directing their anger at the players.
But when he was informed the trash was aimed at the referees, Prescott's message changed: "Credit to them."
However, on Tuesday, Prescott released a statement on Twitter as he issued a fulsome apology.
"I deeply regret the comments I made regarding the officials after the game on Sunday," Prescott said. "I was caught up in the emotion of a disappointing loss and my words were uncalled for and unfair.
"I hold the NFL Officials in the highest regard and have always respected their professionalism and the difficulty of their jobs. The safety of everyone who attends a game or participates on the field of a sporting event is a very serious matter.
"That was a mistake on my behalf, and I am sorry."
CNN has reached out to the NFL for comment. However one US sports refereeing body has already called on the NFL to sanction Prescott.
"The NBRA condemns the comments by Dak Prescott condoning violence against game officials," the NBA Referees Association posted on Twitter.
"As an NFL leader, he should know better. We encourage the NFL to take action to discourage this deplorable behavior in the future."
On Monday, Cowboys executive vice president Stephen Jones did criticize those fans who had thrown debris.
"That's just unfortunate," Jones told 105.3 The Fan on Monday. "That's not the way I see our fans. I think we are a class act. There is just no place for that."