Born in Crediton, Devon, Sam was in Plymouth Argyle's youth academy before Southampton snapped him up in April 2012. He made his debut seven months later and went on to appear 18 times in the Premier League, scoring once. He was recalled from a six-month loan spell at MK Dons in January 2016 and then moved onto Blackburn Rovers on loan the following season. Gallagher impressed in a struggling Rovers side that got relegated, scoring 13 times in 42 Championship appearances.
Sam signed a new four-year contract with the Saints in July 2017, before returning back out on loan for more competitive football to Championship side Birmingham City. During his loan season at The Blues Sam became the team's top league goal scorer. Gallagher has played international football, representing Scotland at Under-19 level before switching to England in 2014.
In July 2019 Sam signed a permanent four year deal with Blackburn Rovers.