I did this a few years ago coming in on BLR-DXB (separate ticket) and departing on UA DXB-IAD - I had the same question as you, so I checked in online just in case the re-security check asked me for an onward BP to re-enter the terminal. They never even asked for it, but probably a good idea since I don't think UA has a transfer desk in T3.

You can easily walk from Terminal 3 to Terminal 1 (as someone else said, it's all linear).