A thunderbolt from Gaku Shibasaki had seen Getafe lead in the latter stages of the first period, but Denis Suarez and Paulinho both came off the Barcelona bench to turn it around for the league leaders.
The goal was Paulinho's first since joining the Catalan outfit from Chinese Super League side Guangzhou Evergrande last month, and the Brazilian was glad to prove a point to his doubters.
"I'm very happy about scoring the winning goal. The most important thing is that the team got back on the scoreboard, which allowed me to score comeback goal," Paulinho told reporters.
"Anger in my celebration? It was a celebration after a difficult match. Ultimately, the most important thing was to come out with the three points. It's difficult to score in these situations, but I received a great pass from [Lionel] Messi and I'm very happy for my goal."
Paulinho's move was questioned in various quarters after the midfielder struggled to impress at Tottenham Hotspur between 2013 and 2015.