The best chance of the opening 45 minutes looked to have fallen to Hendrick as he surged into the box to latch on to a long pass forward, but the move was thwarted by the offside flag.
Barnes and Hendrick both mustered decent chances around the hour mark and Burnley's pressure was soon rewarded with the opening goal.
Aaron Lennon's cross into the middle was volleyed towards goal by Gudmundsson and kept out by McCarthy, only for the ball to fall to Hendrick, who flicked on for Barnes to nod home before the Saints stopper could get to it.
Southampton set about trying to get back into the game and came close on 75 minutes when Carrillo's threaded ball was picked up by Josh Sims, who forced an acrobatic save from Pope with a curling effort towards goal.
Just as Burnley looked to get their first win of 2018, Italy international Gabbiadini lashed home from Carrillo's delivery to finish off a move during which referee Bobby Madley had inadvertently impeded Ashley Westwood.
Southampton climb to 16th in the table thanks to their point, while Burnley remain seventh in the standings.
BURNLEY (4-4-1-1): Pope; Lowton, Long, Mee, Ward; Lennon, Cork, Westwood, Gudmundsson; Hendrick; Barnes
SOUTHAMPTON (4-2-3-1): McCarthy; Cedric, Stephens, Hoedt, Bertrand; Romeu (Boufal 75'), Lemina; Ward-Prowse (Sims 68'), Tadic (Gabbiadini 81'), Redmond; Carrillo