Puzzling! I had always imagined an activation energy as positive.
I sniff a hidden question in the question:
The produced energy does not need to exceed the activation energy, because the reaction energy is measured between the initial and the final state, not from the energy peak.
And: you don't need the products to provide more than the activation energy to the reactants, because heat is a statistical value. If the activation energy is 1000K, the reaction can proceed at 300K - just at some lucky molecules at a time. Additionally, this energy can be provided by the initial temperature.
Think of an equilibrium: even with an activation energy, both reaction directions (hence H>0 and H<0) will proceed provided the hill isn't too high.
But a relation with the initial conditions, sure.