The union's motion is in response to a ruling last week by a federal appeals court, which lifted an injunction that prevented Elliott from serving a six-game suspension over an alleged domestic-violence incident.
The 5th U.S. Circuit Court of Appeals in New Orleans on Thursday voted 2-1 in favor of the NFL, which requested an emergency hearing after a district court in Texas blocked the suspension last month.
The appeals court ordered the Texas court to dismiss Elliott's case.
The union acknowledged Monday morning that Elliott's suspension was in effect and the second-year running back was not allowed at the team's facilities.
Elliott will not be allowed to practice or play and cannot have contact with any of the team's coaches while the suspension remains in place.
On Friday night, the NFLPA requested a hearing of the full panel of judges with the 5th U.S. Circuit Court of Appeals in New Orleans to reverse the previous day's decision.
With no action forthcoming by the full appeals court the NFLPA's request, the union filed its motion Monday to seek relief in the Southern District of New York.
"The Texas Action is now in a procedural limbo such that the NFLPA must seek preliminary injunctive relief before this Court," the NFLPA wrote Monday.
Elliott received a six-game suspension from the NFL on Aug. 11 when he was found to be in violation of the league's personal-conduct policy. Elliott, who was not arrested nor charged in the case, maintains his innocence after being accused of assaulting his former girlfriend, Tiffany Thompson.
The original punishment was issued by NFL commissioner Roger Goodell after a 13-month league investigation.
The ban was upheld by arbitrator Harold Henderson on Sept. 6, but Elliott received a restraining order two days later allowing him to continue to play.
The NFL responded by filing an emergency motion with the U.S. Court of Appeals for the Fifth Circuit to enforce the suspension, but a U.S. District Court judge in Texas denied the league's appeal for a stay of the preliminary injunction.
On Thursday, Pratik Shah, a lawyer representing the NFL, insisted the NFLPA filed its case in the wrong jurisdiction per last month's hearing in Sherman, Texas.
The Circuit Court agreed and said in its majority decision that the Players Association filed its lawsuit to have Elliott's suspension dissolved before an arbitrator could make his decision.
The Cowboys had a bye last week. They are schedule to play the 49ers on Sunday at San Francisco.
Elliott has rushed for 393 yards on 105 carries in five games while his case was in the court system. As a rookie last season, he led the NFL with 1,631 rushing yards.