The commissioner's office said Tuesday that Archer has appealed his penalty and will play pending a decision. Puig and Bell were to serve their discipline starting Tuesday.
Chief Baseball Officer Joe Torre also fined all three for their behavior Sunday.
Cincinnati's Derek Dietrich stood in the batter's box to admire his long home run in the second inning, and two innings later Archer threw his first pitch behind the waist of Dietrich. Plate umpire Jeff Kellogg warned both benches and Bell jogged from the dugout to the infield, arguing Archer should have been ejected.
Benches and bullpens emptied, and an enraged Puig barreled into a bunch of Pirates.