Just won on Easy in a Gigantic or Huge (don't remember which) galaxy with 4 opponents.
I apparently won an Influence victory. But I'm not quite sure how. Many of the AI planets that flipped has a larger influence points level than my planets did, but I still flipped them. Not sure how this works.
Some things I did:
- Kept onside with the strongest factions/malevolent factions by building embassies on every planet and regularly declaring my friendship
- Played off the strongest factions against each other - made alliances with the strongest and bribed them to attack the other AI
- I kept stable and patient and played my game, despite being much weaker than most of the other factions for much of the game
When victory came, it came quickly. Two major AI factions surrendered to another AI faction and gave their assets to me, as I had been kind to them in the past. This quadrupled my power.
Then I culture flipped the Krynn, not sure how though. Finally, despite being stronger than me, the Arceans very quickly crumbled under the influence of my planets. They wanted to be like me 
I beat them all, despite having by far the weakest military throughout the game. So the AI goes mad building up massive militaries but they can still lose.
Next game I will play on Normal and see how I do.