This isn't as hard as it may first appear, but there is a method to make it easier than just being careful.
Firstly, you want to get the utility crate in the bottom left of the map ASAP. Use jet packs to get over there and grab it, as it contains telekinesis.
With that same worm after using the teleport gun to get close to the UFO, use telekinesis on the UFO and drop it down into the hole.
You can now safely shoot/let the barrels explode. Just be mindful of the flames and where they will fall, as they can fall down the hole onto it.
I personally didn't explode the middle set just in case, but did explode the two side sets from underneath to wipe out the above worms quicker.
If you don't get telekinesis early on, do not stand near any barrels as enemy worms will blow them damaging the UFO if it isn't moved.
Once it's moved, wipe the enemy worms out and the achievement will pop upon level completion.