If an invasive species preys on native species, spreads widely and becomes dominant, should it not become extinct soon enough because of a lack of food?
This potentially depends on what species you are referring to.
In the case of plants, photosynthesis is used to as their source of energy, so will never run out of food as such. Assuming the conditions are favourable in terms of water and temperature (likely, otherwise the species wouldn't have flourished in the first place) and they stay that way, the only limiting factor is going to be nutrients in the soil. These are returned to the soil when the plant dies, so this is fairly self-limiting, when there's not enough then plants will die and the nutrients will return to the soil.
Animals face a slightly different situation, in that they need to eat to survive. As such, if the population that was introduced was large enough to eat all the prey (whether plants or other animals) in the area and carnivorous animals didn't partake in cannibalism even when times were tough, then the animal may die out.
In general however, predator and prey regulate fairly effectively, a decrease in prey causes a decrease in food availability which leads to a decline in predators. This allows an increase in prey, which gives a higher availability of food to predators so their numbers increase. This causes a decrease in prey numbers and so on. In general once an invasive species has been established, it will require intervention for it to be removed and it will not die out naturally.