Typhon had not been born when The First Titan War had taken place however he was the greatest threat The Olympians had ever faced and they (with help from some gods who were not Olympian) were barely able to defeat him. However, in the end they did and Zeus buried him Under Mt. Etna.
Just as Olympian culture moves based on where the most prominent civilizations are at the time so did Typhon and where he was buried and this time it was under Mount Saint Helens. Typhon however like so many monsters do escaped his prison.
When Typhon escaped, the vast majority of The Olympians went out to fight him. Demeter however was in The Underworld with Hades and Persephone and Poseidon was busy fighting Oceanus. The battle lasted for days and caused massive destruction of cities and states. Typhon's intent was to make his way north to Manhattan to help destroy Olympus and Camp Half-Blood. Dioysius had been hit out of the sky and landed somewhere in the Appalachians buried under rubble in an abandoned coal mine while Hephaestus had been similarly immobilized landing in West Virginia so hard that he created a new lake. Percy ended up going to Mt. Olympus, and by sitting on Poseidon's throne attracted his father's attention he told his father that he had to go help the other Olympians. While Poseidon did protest somewhat claiming Oceanus would destroy his palace, Percy convinced him that the Olympians could not defeat Typhon without him and without the Olympians Kronos would win anyway. It was a short time later that Typhon made it to New York where the Olympians were making their last stand when Poseidon arrived fully refreshed with a battalion of thirty foot tall Cyclops as well as Briares all riding huge sharks, dragons and sea horses. The Cyclops were carrying anchors and chains that could be used to hold Battleships. The battalion of warriors used these to wrap Typhon's arms and legs weighing Typhon down while the Olympians attacked with renewed effort. Eventually, Poseidon caused the water in the Hudson to rise and used it like a cocoon dragging Typhon down. Poseidon created a special tunnel at the bottom of the river for Typhon leading straight to Tartarus.
The Battle against Typhon was one fought by most of the Olympian themselves against Typhon. It was part of a three pronged attack strategy by Kronos. Kronos had Oceanus launch a siege on Atlantis keeping Poseidon and the Atlantian Army occupied so he would not be able to help the other Olympians. Kronos was also counting on the discord between Hades and his brothers as well as Demeter's desire to remain close to her daughter that they would not get involved. With no gods of great power near Manhattan his plan was to Attack Manhattan with the majority of his army and destroy Olympus and whatever defenders it had. Kronos however overestimated Hades' disdain towards his brothers as well as the importance Poseidon's Palace was. Kronos further underestimated how well the demigods and their allies in Manhattan would do against the Titan Army.