The decisive move in Gaugamela was Alexander's decision to hit the center of the Persian line which was currently engaged by the phalanx. He successfully outmaneuvered the Persian cavalry from their left flank and had his Hypastists/lighter infantry and cavalry draw them away from the battle. This allowed Alexander to filter his Companion cavalry through the gap and hit the weakened center where Darius was. Its believed that Alexander's whole gameplan was to cause enough chaos for Darius to flee (as he was known to do that). Alexander was of course correct in his assumption, and Darius routed shortly afterward with most of the Persian center tailing behind. It is true that most of the heavy lifting was done by Parmenion and Antigonus, who held the center and left phalanxes together, but it is highly likely that without Alexander's tactics, the Macedonians would have been encircled and destroyed.
The phalanx is unique in warfare for its ability to pin nearly any enemy with its pikes/spears and hold them till said enemy is either slowly cut down or outflanked and crushed by another group. Alexander utilized this to the greatest extent possible in nearly ever battle he fought in. In fact, by the time Alexander had routed the Persian main body at Gaugamela, he decided to head back towards his left flank, which had nearly been overrun by Bactrian cavalry and infantry, and subsequently saved Parmenion by smashing the remaining Persian's flanks. This wasn't the only time Alexander had to save Parmenion's ass. During The Battle of the Granicus, the phalanx had been caught up by rough terrain and their tightly packed formation was nearly ruined. Alexander had to kill the Persian leader and his lieutenants on the other side of the river to even salvage the chance of victory, which he did. Let's just say that Alexander and Parmenion had to rely on each other for victory. Alexander needed the infantry to hold so that he could break the rest of the opposing army, while Parmenion's whole survival was dependent on the decisiveness of the Companions. In most cases, it does come down to subjectiveness, as most records about Alexander are rather scarce and blunt. Some say he was a tactical genius who conquered most of the known world, while others believe he was an egotistical, arrogant, drunken buffoon driven by a mad belief that he was a god.