Asham has been banned four games for cross-checking Philadelphia's Brayden Schenn while Shaw was suspended for three games for hitting Phoenix Coyotes goalie Mike Smith.
Shaw began serving his suspension on Tuesday when the Blackhawks resumed play against Phoenix while Asham's suspension begins on Wednesday when the Penguins play Game Four in Philadelphia.
The first round of the play-offs have been characterised by outbursts of fighting and physical play, particularly in Game Three of the series between Pittsburgh and Philadelphia.
The game featured 133 penalty minutes and first period ejections of Pittsburgh's Kris Letang and Asham and Philadelphia's Kimmo Timonen.
Pittsburgh's James Neal was banned one game for charging Philadelphia's Claude Giroux in Game Three while Washington's Nicklas Backstrom received a one-game penalty for cross-checking Boston's Rich Peverley in Game Three of their series.
The Bruins and Capitals play the fourth game of their series on Thursday in Washington.