Did an advanced search on Twitter for specifically "Dawgs win again" from 2011 to Feb 2021 and again starting with Feb 2021 to present day. There are sporadic mentions of that phrase (and most are not Mississippi State related) before the 2021 season, but starting in March 2021, it's 99% Mississippi State fans Quote Tweeting baseball wins, so I would say as a catchphrase, it's really only been last season.
I think the first time I heard it take hold was Riley Self whispering it into the camera on the little videos the baseball team would put out after weekend series wins last season.