The Giants went down 27-26 to the unbeaten Patriots as the hosts converted a last-gasp field goal after what was initially awarded as a Beckham Jr touchdown was overturned with two minutes to play.
The 23-year-old was unable to hold an Eli Manning pass in the endzone which would have put his side clear in front in the closing stages, and he admitted his frustration at failing to seal victory for his side.
"I lost us the game with the play down in the endzone, a play that should have been made," he told reporters after the game. "You cannot leave it up to the officials to get anything right. You have got to make the play yourself and it was just a case of playing the play longer than the opponent.
"I caught it. I trust in my hands. I go up and catch it. You come down with two feet and you cannot try to do anything too early. You have got to complete the whole process. It did not go the way that we wanted it to."
The defeat has left the Giants 5-5 to start the season, although they still sit top of the NFC East after 10 games.