The win was compensation for the Charlie Hills-trained horse, who was narrowly beaten to the recent 1,000 Guineas at Newmarket by Sky Lantern.
Just The Judge started as 2-1 favourite with Jamie Spencer on board, and beat 40-1 outsider Rehn's Nest by a length and a half, with Just Pretending (14-1) third.
Hills' father Barry masterminded wins in the race twice, and Sunday's winning trainer was delighted that his horse had let him follow in his father's footsteps.
"She's a diamond filly," said Hills. "She's a great filly to deal with, so easy and very professional. Jamie has given her a peach (of a ride). He rode her just like he said in the paddock to the stride."
Spencer was also thrilled: "She's a smart filly. I got a good position and at halfway I was very happy and it all went really well She's a filly we always thought would want further and she'll be even better when she goes further."
Winning owner Sheikh Fahad believes that the horse can go on to even bigger success: "We were very confident. She has been a lot better in her coat and hopefully she can go on improving.
"This is the first proper season for Qatar Racing so to win a Classic is a great start."