The cartoon's title is a play on Tugboat Annie, and is the only cartoon in the Warner Bros. series to bear Granny's name.
In the opening scene, Granny and Tweety are happily piloting a rented tugboat in a harbor, singing a kiddie-song duet about the carefree joys of their activity. It is the only scene in which Granny appears, as the rest of the cartoon is devoted to Sylvester's latest attempts to catch and eat Tweety, which begins when, after failing to grabg a fish by hiding in a fisherman's basket, and gets attacked by a crab) he sees the boat carrying the canary chug past.
Sylvester's attempts, all unsuccessful, include the following:
Using a rowboat to get to the tug. Tweety drops anchor in the boat's hull, sinking Sylvester (though he comes out onto the beach still rowing with the oars).
Using an inflatable raft, which is deflated by Tweety's dart ("Hey, puddy tat! Wook what I found! Here, you can have it!").
Two attempts at jumping off the bridge. The first jump is mistimed, as Sylvester lands in the smokestack, leaving him with his rear end on fire. A follow-up attempt from another bridge to parachute onto the boat's deck ("Oh, that bad ol' puddy tat! He never give up!") results in a jammed pack, which only opens after Sylvester sinks to the bottom of the canal ("Aww, the poor puddy tat. Got himself all soaking wet.").
Using a snorkel to swim to the boat. A seagull finds Sylvester's pipe the perfect resting place (blocking the airway); the cat is left holding the egg.
Driving a motorboat, but Sylvester drives instead into the rapids and over a waterfall.
A lasso ("That puddy tat thinks he a cowboy!"), which instead grabs the antenna of a speedboat. Sylvester decides to show off several waterskiing tricks ... until the inattentive puddy smashes into a pole. As he floats upside-down, a fish gurgles Tweety's signature line: "I tawt I taw a puddy tat."
Some of the scenes would later be recycled in one Pink Panther cartoon, due to the fact that some of the Warner Bros animators would establish DePatie-Freleng Enterprise as well as being hired by United Artists to create Pink Panther and other cartoons 8 years later.