The 29-year-old has scored two goals in four matches since he joined the Pirates on an initial month-long deal at the end of October.
Rovers manager John Ward was delighted Beardsley decided to extend his stay at the Memorial Ground.
"Chris is a really good team player and a senior pro who brings experience to our young team," Ward told the club's official website.
"As well as scoring a couple of goals, he is hard working and has a great attitude, both on and off the pitch. He has done everything I hoped he would do here, and more."
- Sports & Recreation