The reason that believing in Jesus specifically is the only route to salvation is due to not accepting his sacrifice and therefore being stuck with original sin, while believing in him and accepting that sacrifice gives you another chance.
Original sin's kind of an interesting thing, mostly because it's not necessarily
biblically supported; there's room in the text for the interpretation, iirc, but nothing particularly explicit. You can blame most of its dogmatic existence on Catholicism in general and Augustine in particular.
As I understand it, insofar as original sin goes, even if you
do accept its original existence (and there's plenty of ground not to), at this point it's utterly irrelevant regardless as to if you're a believer or not. It's one of the reasons Jesus was supposed to be as awesome as he's considered to be and his sacrifice as important; when he did his thing, he absolved pretty much all sins of that sort, for
everyone, period. Christ was by and large (though, as in all cases with a body of text of the nature of the Bible, there's exceptions) a pretty decent fellow.
If you're trying to take the high road with things (it's fair to note that many, often especially those claiming Christian belief,
don't.), "Jesus as the only route to salvation" is closer to Buddhist practice than anything -- a practical rule and action set for obtaining happiness in this life and the next. The path to salvation is to walk
as Jesus; to act as he acted, to love as he loved, etc., so forth, so on. This is a thing guaranteed, the belief system says, to bring goodness (happiness, an improved situation for yourself and those around you, a reduction of harmful acts by yourself and those you interact with, etc.) into your life. The metaphysical beliefs inherent to that are (or were; it's incredibly important to know that the people who wrote the bible had a
wildly different perception of reality and metaphysics), so far as I can tell, supposed to be necessary consequences of doing so.
Personally, I think a lot of people nowadays strongly overstate the belief aspect ([cynic]often, it seems, as a method to cover their own asses and believe themselves to be achieving salvation despite not actually doing shit to achieve it [/cynic]) of Christian theology. I've been fairly well convinced for a while now that if Jesus came across an atheist, or an agnostic, or someone of wildly different theological precepts, so long as those people were working the good works and generally being awesome to one another, Jesus would distribute righteous brofists and probably chip in with whatever was going down.