Let me point out that they did conquor Elsweyr through deception. Be it through knowing what would happen with the void nights or by making it happen. Claiming they fixed such a thing with little proof seems far fetched to me.
The only people with such power to be able to make both moons disappear would be the Psijic Order or the Aedra.And how did they 'save' the race?
Cause, the way I see it the Aldmeri Dominion would not have lasted an assault on Elsweyr without suffering severe casualties. Fair enough they'd have an easy road to Torval since its so close to the Valenwood Border. However the deserts and tribes would make any sort of unified Army marching through there to be quite literally one massive target that would suffer from dehydration, heat exhaustion and constant hit and run tactics from Tribal Warlords. Much like the Empire instead they used a massive robot. Lets not forget what they did to the area around Halls of Collossus (Empire
)
So the elves would have either A: Used some form of divination from the collective power of Crystal Tower or B: An Orrery and some how either predicted that the moons would disappear or they in a rather unlikely scenario managed to summon up enough power to actually black out the moons for five years. Last bit doesn't seem feaseable with me much though.
Therefore, we know the Dominion could not have taken out the moons. However, the long-lived High Elves of the Thalmor do have the knowledge and wisdom to figure out what was going on and restore them. It's not like Mortals haven't been able to undo cosmic catastrophes in the past.
The Aldmeri Dominion didn't have any intention to invade Elsweyr - Annequine and Pelletine revolted against Imperial misrule, dismissal, oppression and subjugation, and their re-formed, long-time allies in the Aldmeri Dominion (Look at Elsweyr's history - As long as there's been a Dominion, the Khajiit have been strong allies with them) offered them the strength and protection to rebel against the Empire. Furthermore, the Khajiit of Elsweyr have had more freedom of self-determination as a protectorate of the Dominion than they did as a Province of the Empire. The Dominion has also been able to ease-if-not-eliminate hundreds of years of racial tensions between the Bosmer and Khajiit. (Tashiir loves vactationing in Valenwood - the local style of dress particularly appeals to her, especially on the younger Bosmeri men.)
They are also no longer thieves and smugglers within the Dominion - a role they were forced to mantle and idealize under Imperial Oppression, as it gave them a way to "fight back" without going head-to-head with the occupying army.
When the Thalmor came, it was with salvation, protection against Elsweyr's enemies, and humanitarian relief for the Khajiit.