Alexander, for the most part, left things unchanged in the lands he conquered. He didn't impose Greek customs, respected (or perhaps ignored) local religions and cultures and allowed a certain degree of self government that, for several of the territories of the former Achaemenid empire, was quite a refreshing change. Not everyone under his rule accepted him, but most did, and several satraps capitulated without a fight. The Egyptians went a step further and proclaimed him a son of Amun. Several of his opponents even kept their positions (or attained other positions of power) in the new empire (the more notable example being Porus).
Of course, Alexander would leave a garrison here and there, and would raze cities to the ground when he felt they could oppose them in the future (see Tyre & Gaza), but his main methods of controlling his vast territories were the image of liberator he cultivated, and extremely tight financial control. Although the former satrapies maintained their lands and a higher degree of autonomy than they enjoyed under Persian rule, Alexander denied them any control over taxation. The vast network of cities Alexander founded during this time ensured the Greeks also had a quite strong grip on the empire's trade.
That said, Alexander's empire didn't last long. It splitted into several states almost right after his death and the conflicts between his successors went on for almost half a century. These states, once normalized, were much smaller in comparison and thus easier to control. In general, the successors continued Alexander's plan: they avoided antagonizing the local population, kept tight financial control and continued expanding the already vast network of cities Alexander had build.