cosmology - How did the universe shift from "dark matter dominated" to "dark energy dominated"?

In order to get dark energy to dominate, wouldn't you first need another form of energy to push the expansion until dark energy could dominate? Otherwise I don't understand how the universe could shift from having a decelerating expansion to an accelerating expansion. Is there any analogy that could help understand this?

Answer

The title and the text actually ask two different questions. While Kyle Oman and Thriveth answer the title excellently, I'll address the question in the text which asks "Why did the Universe expand in the first place, before dark energy (DE) started to dominate".

The answer to this is inflation (we think). The first fraction of a second after the creation of space, it was dominated by "something" that mimicked the effect of DE, causing space to expand by a factor of $\sim e^{60}$. The epoch of inflation lasted until the Universe was some $10^{-32}\,\mathrm{s}$ old.

The expansion continued, but were slowed down by the mutual attraction of radiation, and later matter. If the ratio of DE-to-matter had been smaller, this attraction might have slowed it down sufficiently to halt the expansion before DE started dominating, but that was just not the case in our Universe.

Now what caused the inflation is another question, which someone else than me is better at answering. But I think the most accepted theory, or rather hypothesis, is some scalar field consisting of inflatons.

Analogy

You request an analogy. I can give you the following:

Throw a rock into the air. Your push is inflation. The distance from Earth to the rock is the size of the Universe. The gravitational force between Earth and the rock is the mutual attraction between various forms of energy in the Universe. The speed of the rock is the expansion rate of the Universe. Now if your pitch was too weak, the rock will eventually fall back (Big Crunch), while if you throw hard enough (11 km/s), the rock will escape Earth's pull (Big Freeze). But even if the initial speed was less than 11 km/s, if the rock comes sufficiently close to the Moon (dark energy), it will start picking up speed and eventually escape.