I had exactly the same thing happen with my daughter from 4 months to 6 months. Most frustrating thing ever. We used the No Cry Sleep Solution (I mention it a lot on here, but I swear I'm not a paid advertiser or something... it just worked for us).
I also started giving her 1-2 minutes when she woke up at night to see if she was really "awake" or whether she was just coming out of REM sleep and would settle herself back down. I found that she started gradually dropping one of her every two hour wakings per week or so until she was finally sleeping through the night. I think the 4am one was the last to go, so it seemed to work from early night wakings being eliminated to early morning being last. Which makes sense, since she'd be legitimately hungry by that point (at least that was my thinking).
Anyway, she still at 13 months will occasionally cry out in the middle of the night and then settle herself back down. I just wait long enough to judge the cry... if it's ramping up and getting upset, I go in. If she's sort of complaining after her first cry, I see whether she'll just settle herself. That worked really well for us starting at around your baby's age, which is, incidentally around the time they develop self-soothing abilities. Good luck!