Cosmic inflation changed the expansion rate of the "explosion", but this doesn't mean there wasn't an explosion.
Cosmic inflation changed the density ratio between spacetime and mass/energy, but the entirety of the universe's mass/energy came into existence at the moment of the Big Bang. So yes, some explosion that.
Remember about this particular explosion that it wasn't an explosion of mass/energy into the empty space of a pre-existing universe, but an explosion of mass/energy and spacetime into nothing -- the Big Bang was space, time and mass/energy.
In other words, there was no "before" before the Big Bang, and there was no "outside" outside the Big Bang. The Big Bang was everything.
Cosmic inflation changed the density ratio between spacetime and mass/energy, but the entirety of the universe's mass/energy came into existence at the moment of the Big Bang. So yes, some explosion that.
Remember about this particular explosion that it wasn't an explosion of mass/energy into the empty space of a pre-existing universe, but an explosion of mass/energy and spacetime into nothing -- the Big Bang was space, time and mass/energy.
In other words, there was no "before" before the Big Bang, and there was no "outside" outside the Big Bang. The Big Bang was everything.
More detail here: http://arachnoid.com/gravity