You lost because of bad generalship, not because of ASF.
Against High Elves, that's always going to be the case.
Always Strikes First makes the army playable. If you look at the points per model for any given High Elf, you'll notice that they're almost invariably around or above 15 apiece, and they're T3, S3, and at best wearing heavy armor with no shield. In other words, they're very, very, very squishy. The only way that a small army that's so squishy could possibly be playable is if they get their licks in first, to offset either the return attacks making a mess of them, and help offset the fact that they're going to get outmaneuvered by more numerous armies.
Always Strikes First is really all the army has going for it, as far as special abilities go. Every army has something-- Wood Elves move and shoot without penalty, Dark Elves hate everybody, Orks have Choppas and the Waaagh!, Chaos re-rolls failed panic tests and has Eye of the Gods... I mean, in the grand scheme of things, Always Strikes First is pretty even as an army-wide ability goes.
If you can somehow take away ASF from the High Elves, you're going to roll them like a cuban cigar-- off the top of my head, there's only the ability the Herald of Nurgle can buy, but I could easily have missed something. If you also have a character that always strikes first, you'll probably get the jump on the High Elves, to boot-- it's an increasingly common magic item trait, it seems, and most characters that can take weapons like that have at least comparable initiative to the High Elves' elite troops.
The other option is to plan to be defensive against the High Elves. Rather than try to exploit the fact that they're T3 and S3, try and make your guys tough enough by buying shields and armor if they don't come with it, and try to weather the attacks from the Elves so you can return it in spades. A unit of Dwarf Warriors, for example, can be given great weapons, but they also come with hand weapons and shields. Normally, giving them great weapons is a good call, but if you consider that a Dwarf Warrior has a 3+ armor save against S3 attacks in close combat, they can weather the 15 attacks that the High Elves' spearmen are going to throw at them. The return attacks will probably kill more elves than lost dwarfs, so it comes down to static resolution. When you factor in static resolution, the Elves are in bad shape, because there just aren't that many of them.