"It's not acceptable and it's something I gotta figure out," Elliott said postgame, according to Jon Machota of The Athletic.
Elliott lost two fumbles against the Cardinals and rushed for just 49 yards. The three-time Pro Bowler has now lost five fumbles through six contests this year.
Overall, the Cowboys were held to a season-low 10 points Monday and committed four turnovers in their first game since quarterback Dak Prescott's season-ending injury last week.
"I don't think we can use that as an excuse," Elliott said of Prescott's absence, per ESPN's Todd Archer. "I'm just going to keep saying it over and over - I started the game out with two fumbles, gave the ball away, and gave them all the momentum they need to go take off. I want to say I'm sorry, and this one is on me. I need to be better."
Dallas head coach Mike McCarthy also lamented his team's turnovers postgame.
"We're the worst in the league at taking care of the football," McCarthy said, according to Michael Gehlken of the Dallas Morning News. "It's frustrating to see the pattern here in five of the six games this year."
The 2-4 Cowboys have now turned the ball over 15 times this season, the most in the NFL through six weeks.