Both also were fined undisclosed amounts for their conduct during the incident. The announcement was made Monday by Bob Watson, vice president of on-field operations for Major League Baseball.
A Cubs spokesperson said Lee was appealing the penalty.
Both suspensions were to begin Tuesday, when the Cubs play at Texas and the Padres play host to the Baltimore Orioles.
Cubs hitting coach Gerald Perry received a three-game suspension and undisclosed fine for his "aggressive actions" during the incident. His suspension will begin Tuesday.
In addition, Padres players Brian Giles and Jake Peavy received undisclosed fines for their roles.
Lee was upset at a pitch that Young threw in the fourth inning Saturday that was high and close to his head. It nicked his left hand. As Lee walked to first, he said something to Young, and the pitcher responded. Lee didn't like Young's comment, and took a swing. Young swung, too, and neither landed a punch but both benches emptied and play was delayed several minutes.
"We had some dialogue, but that's between us," Lee said Saturday. "I don't get upset too often, but I don't like when a ball is thrown at my head. It was thrown right at my head."
Perry wasn't sure why he was tossed. Peavy had jumped on Lee's back, and Perry was trying to pull the Padres pitcher off the first baseman.
"I was as shocked as anybody that I was thrown out of the game," Perry said Saturday.
The Cubs were traveling to Texas on Monday, and unavailable for comment on the penalties.
Carrie Muskat is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.