The 19-year-old has linked up with the League Two side on a temporary basis and will spend the remainder of the 2021/22 campaign at the Memorial Stadium.
Anderson - whose grandfather Geoff Allen played for the Magpies between 1964 and 1968 - has been on the fringes of the United first team this season, having made both his Premier League and FA Cup debuts against Arsenal a year ago.
He has been in fine form for Newcastle's under-23s, though, scoring seven goals in nine outings across all competitions for Elliott Dickman's side.
The switch sees Anderson, who has been capped by both England and Scotland at youth level, link up with former Magpie Joey Barton, the current manager of the Gas.
Rovers currently occupy 12th spot in the League Two table, and Anderson could make his debut for Barton's men on Saturday afternoon when they visit promotion-chasing Sutton United.