When you hear that "the U.S. birth rate is near or below replacement rate", what is probably meant is that the total fertility rate (TFR) is below 2.1 (replacement rate).
It is possible that the TFR is below 2.1, while births exceed deaths. One way this can happen is if there's a rising share of women entering reproductive age.
This phenomenon is sometimes called population momentum (more details: 1, 2, 3).
This is actually a fairly common phenomenon: Many countries have had TFR below 2.1 for many years but their populations have continued to grow. For example, China has had TFR below 2.1 since 1992 but births have continued to exceed deaths every year.