Roped into babysitting little Egbert, the child of a friend, Fred and Barney take the tot over to the pal Joe Rockhead's house to watch the fights on television. Egbert befriends Joe's pet runtosaurus and dresses the creature up in his baby clothes, which results in chaos when the pet--whom Fred and Barney mistake for the baby--escapes from the house and dashes up a tree.
Fred is late getting Barney to work. Barney gets called to his boss's office. Barney's boss gives him tickets to the big fight in town. Barney tells Fred when Fred comes to pick him and they are both excited.
Fred gets home and finds out that Wilma has volunteered he and Barney for baby sitting duty the same night because of a bridge club meeting. She makes Fred feel like a heel and he agrees to do it. Barney comes over and has had the same experience with Betty.
The girls head out for their bridge game and the guys turn on the TV to watch the fight and find out that it's been blacked out to within 25 miles of the fight. Fred gets an idea about how they can still watch the fights.
They go to their friend's (Joe Rockhead) house and bust in to watch the fights since Joe isn't home. They get the baby ,Egbert, to bed and Barney lets Joe's little dinosaur stay with the baby. Egbert takes off his cap and shirt and puts it on the dinosaur. The dinosaur runs out and Fred and Barney think that it's Egbert and chase it.
Joe comes home to find his door busted in and the baby there and Joe assumes the baby broke in and takes it to the police station. He tells the chief the story and gets locked up.
Fred and Barney chase the dinosaur up a tree and try to get it down. A policeman comes by and they tell him that there is a baby in the tree and the policeman calls the fire department to get it down. They find out that it is actually a little dinosaur and the police haul Fred and Barney to jail.
Wilma and Betty see the news report about the baby in the tree and they see that Fred and Barney are arrested and then they see Egbert. They go to the station with Egbert's mother and she takes Egbert home. The girls leave the guys in the jail because Fred says that it would take too long to explain.