No this will not be the case, you have the cause and effect reversed - allow me to explain.
First lets simplify this and not consider causes such as illness, death from accident etc, so we only consider the population of a food chain 'tier' purely from its role within that food chain.
In a situation like this, the population is limited by 2 factors:
1, predation from the tier(s) above it
2, availability of food as tier below it.
If you take away 1 of these limiters, in this case, the predation from above, the population is still limited by that second factor - availability of food.
Take the sharks out from this, the secondary predator species, may experience localised small population increase, but overall, their number would not explode because there simply isn't an infinite amount of food to sustain them. So it's not that a high tier will over populate and wipe out a lower tier but rather they will NOT over populate because they can't - there simply isn't enough resource for them to over populate.
Look at it this way: If there is a forest and say completely isolated and fenced off from outside, there are of course trees, grass, providing food for a population of wild boars, which in turn is predated on by tigers, say the tigers all died off, the boar population will not increase unchecked - simply cause the forest can't sustain an unlimited number of herbivorous. In short term some trees maybe uprooted and chewed to death by the pigs but then a new equilibrium will establish between the number of trees and number of boars.
Of course like I said this is a simplified scenario, in real life there are other factors at play but they are for the most part, peripheral influences. The main limiting cause, is the available food, or the lack of, of a tier below in a food chain. BTW at the most basic level, the plankton level, the 'food' is sun light. that's why a nuclear winter, or meteorite strike that result in dust particles blocking out the sun, is a real concern because it will knock out the base level (foundation) of food chain and every tier above it, will collapse.