How is dense magma able to rise and punch through continental crust?
Your intuition is quite correct: Other factors being equall, a dense magma will not rise through lighter rocks. The exception is if it is under pressure, in which case a fluid magma will escape through 'pressure release' pathways, either laterally or vertically (or in complex geologic structures, in a variety of irregular pathways). Magmas may begin to crystallize at depth. As they do so, the volatile components, such as water, carbon dioxide, halogen acids, and sulphur dioxide, become relatively enriched in the residual magma, where they can build up enormous gas pressures. The most spectacular results are in Plinian-type volcanoes, and in many less dramatic eruptions. Consider, for example, pumaceous lava, which is essentially gas bubbles frozen-in during the process of escape. At depth there are no such bubbles. but the gasses are still there, and still exerting a huge pressure.

Otherwise, a magma may simply rise because of thermal expansion reducing the density to less than that of the host rock. Most likely the magma will rise as a combination of both mechanisms. 

Also, you mention the Great East Africa Rift, which is notable for some extraordinary carbonatite lavas and related carbonatite-silicate hybrids, which are less dense than almost all of the other African crustal rocks.