After a shot was saved by Edouard Mendy the ball bounced out and Hudson-Odoi and Greenwood both challenged for it with the ball appearing to take the slightest touches off both their arms.
Referee Stuart Attwell, who did not give a penalty initially, was instructed to consult a pitchside monitor by the VAR official, but having watched it, surrounded by rival players, he upheld his decision.
Clattenburg, formerly one of the world's top referees, says Attwell's reasoning behind the decision remains a mystery to him.
"I am amazed Manchester United were not awarded a penalty in the first half for the clear handball by Callum Hudson-Odoi," Clattenburg wrote in his Mail column.
"The Chelsea player’s arm was up in an unnatural position and the ball hit his hand.
"VAR Chris Kavanagh thought referee Stuart Attwell had missed a clear penalty so he gave him the opportunity to review it at his pitch-side monitor.
"How Attwell did not then point to the spot, having been shown the same angles that we were, is a mystery."