Following the attack on the the Grey Council's fleet by an Earth expeditionary force, the Council voted to exterminate the race that had killed their leader, Dukhat, who died in the brief exchange. Over the next two years, the Minbari systematically annihilated Earth's military in their inexorable advance through Earth Alliance territory.
The war ended with the Battle of the Line, which was fought above Earth itself. During the battle, the Minbari captured Jeffrey Sinclair for questioning. During his interrogation, he was examined with the Triluminary, a Minbari religious relic reputed to be keyed to the spirit of their prophet, Valen. The Triluminary reacted so strongly that the Minbari leadership concluded Sinclair must contain the spirit of Valen himself, reincarnated as a Human. Confronted with the possibility that Minbari souls were reincarnating into human bodies and that the Minbari were therefore killing themselves, the Minbari leadership ordered an immediate surrender.
With the unexpected surrender of the Minbari at the Battle of the Line, the Earth Alliance became the only civilization to ever "defeat" the Minbari in war. Consequently, they were included among the major powers with full voting rights in the council that was formed as part of the Babylon Project.
By 2259, Earth actually controlled more star systems than the Centauri Republic. Their military recovery was incomplete, but the newer Omega-class destroyers were far more capable warships than the heavy cruisers they replaced.