Jotaro and Dio's confrontation at the end of JoJo's Bizarre Adventure Part 3: Stardust Crusaders is one of epic proportions. Jotaro is at a seemingly insurmountable disadvantage, as the already immortal Dio's Stand, The World, allows him to stop time and move freely while doing so. If that weren't enough, The World possesses the same physical attributes as Jotaro's Star Platinum, neutralizing what to this point had been Jotaro's primary advantage. How did Jotaro defy the odds to defeat Dio?
The Initial Exchange: How Jotaro Figured Out He Could Stop Time
At the start of the fight, Jotaro goads Dio into a test of strength between their Stands, and this moment is perhaps the most important of all. The two Stands are basically evenly matched in terms of physical ability, with Dio's The World edging Jotaro's Star Platinum slightly because of Dio's vampire stat boosts. However, this result does more than just establish the strength of the World to Jotaro, it also gives him hope.
The fact that the two Stands are nearly equal in every capacity physically tips Jotaro off, or at least gives him an inkling that his Stand could actually be capable of stopping time. Jotaro hints at this immediately, literally stating that he and Dio's Stands are "the same type." Jotaro's acquisition of the ability to stop time via Star Platinum is not random, but the result of him testing a hypothesis inspired by the information gathered earlier in the fight.
Mind Games Work On Vampires Too
From blatantly approaching him in spite of the knowledge of his time stop to deliberately blurting out that he and Dio's Stands are the same, from start to finish Jotaro owns real estate in Dio's head. Dio's descent into paranoia is a gradual one though, the result of an accumulation of small defeats at the hands of Jotaro.
For instance, Jotaro's planting of the magnets in Dio's sleeve to emulate moving in stopped time only works because of the aforementioned statement that he and Dio's Stands are the same. The idea of Jotaro being able to stop time is established by Jotaro himself, but not taken seriously by Dio until he reasserts this claim by moving in stopped time using the magnets. The layers of mental damage build even more when Dio becomes overconfident after discovering the magnet trick, only to have a hole punched in his chest by Jotaro, revealing that, magnet or not, he can move in stopped time for a second. In this moment alone, Jotaro builds Dio's confidence by allowing him to discover the magnets only to tear it back down with the surprise attack, outsmarting him twice with both the magnets and his movement.
These small victories allow Jotaro to win the war, leaving a lasting impact on Dio that trickles into the next exchange. The magnet exchange, his own time stop and other moments prove to Dio that Jotaro is a composed threat. This leads to Dio taking precaution after precaution in moments down the line where charging straight ahead would have lead him to victory. To put Jotaro's mental edge into perspective, the sound of a knife-filled and seemingly lifeless Jotaro scratching pavement is enough to make Dio completely abandon killing Polnareff in favor of completely distancing himself to reevaluate the situation. He even uses a policeman's gun to shoot Jotaro from a distance to ensure that he's really dead, refusing to even get close to the teenager.
10 Seconds is Your Limit: How Jotaro Won Against DIO
After drinking Joseph Joestar's blood, Dio reaches his final form, DIO, allowing him to stop time for 10 seconds. To kill Jotaro for good, DIO activates his time stop and flies off. Despite being frozen in time, Jotaro doesn't panic and waits to see which course of action DIO takes. DIO returns and slams a steamroller on top of Jotaro at the seventh second of stopped time. Here Jotaro stops time for one second in order to break the steamroller apart before freezing again at the eigthth second.
DIO relishes in his victory as he counts down the last two seconds, but at the 10th his body freezes. Jotaro appears behind him, revealing that at the ninth second, he activated his own time stop and escaped the steamroller's impact. DIO's body is effectively frozen in time, as there is an interval between time stops in which the user is incapable of reactivating the ability. From here Jotaro punches DIO's lights out, but doesn't kill him as it would "leave a bad taste" in his mouth.
The final standoff in which the impact of Jotaro's punch to DIO's leg leads to DIO completely crumbling is likely an artistic decision by Araki, but the the rest of the fight and its outcome though should be credited to Jotaro's cunning and composure.