Can I bring a homemade Italian tomato sauce through the US customs?

Not very likely.

The main concern - and your success - would be to convince the Customs that the jar contains what you say it does, and that it is prepared properly (i.e. there are no disease cultures left). Note that they have no means to verify that it indeed contains no meat products (such as bullion), eggs or other prohibited products, and you're unlikely to have a certificate of origin. Here is a list of food products allowed and prohibited from bringing into the USA. As you can see, the list of prohibited products lists much more than just meat products (for example, some sorts of cheese from some countries are not permitted). While canned food are generally allowed, it is unlikely your jar would be considered "canned".

However you can bring any kind of food into the US, and as long as you declare it to Customs, there is no penalty. Then the Customs decide whether it could be passed or not - and if not, they simply confiscate it. Thus if you already made the sauce, or it is not huge investment, you're not risking much.