The origin of continental crust is from the differentiation of Earth, where silicates rose to form the mantle and even lighter silicates rose, then cooled, to form the crust of early cratons.

If the mantle that forms oceanic crust is composed of dense silicates, how is it able to rise up and punch through the continental crust, pushing it apart to form ocean basins, as in areas of rifting or mantle plumes?

My intuition tells me a dense material should not be able to rise from depth and displace a lighter one.

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.

If the mantle that forms oceanic crust is composed of dense silicates, how is it able to rise up and punch through the continental crust, pushing it apart to form ocean basins, as in areas of rifting or mantle plumes?

That is because basaltic magma is less dense than basaltic solid rocks. Basaltic magma may very well be less dense than any silica rich rock it encounters. Furthermore, as Gordon mentioned, there is the issue of overpressure.

Mantle plumes are mostly solid - these are areas that are hotter than their surroundings but are still solid. However, hotter material expands and becomes less dense. This allows the lighter rocks, which are chemically similar to their surroundings, to flow upwards. Melting occurs only at a later stage, either because you depressurise the mantle rocks at shallower depths or because the mantle plume heats up other crustal rocks which melt more easily.