According to multiple recent news reports and articles, this is an accurate claim.
According to the US News article:
> UPDATE, May 26, 2017: Randy Noller, a spokesman for the Department of Veterans Affairs confirmed Irene Triplett is still alive.
[...]
> Irene Triplett – the 86-year-old daughter of a Civil War veteran – collects $73.13 each month from her father's military pension.