Three of the four goals came from penalties, with Gerard Pique opening the scoring from the spot in the 21st minute.
Rey Manaj made it 2-0 three minutes later, following up his midweek hat-trick against Gimnastic de Tarragona with a finish from close range.
Girona pulled one back just before halftime when Samu Saiz scored a penalty before Memphis added Barca's third, also from the spot, late on. Memphis arrived in Barcelona earlier this week after signing on a free transfer from Lyon in June.
He started on the bench but his debut came earlier than expected before half-time when Manaj was forced off with a head injury suffered in the 35th minute. There was palpable excitement among the nearly 3,000 crowd at the game every time Memphis got on the ball.
The Dutchman eventually gave the fans what they wanted, firing a penalty past visiting goalkeeper Juan Carlos with five minutes to play.
"It's incredible," Memphis said after the game. "I'm really happy to be wearing this shirt and to get my first minutes for the club. I enjoyed it so much. The fans were great with their energy and I loved seeing them in the stadium. I am so happy.
"I know everyone expects a lot at Barca. I'm going to work hard and I hope to have great moments with these supporters. I play football to entertain the fans. I hope to bring a lot of creativity. I always try to find the space between the lines and behind the defence, to score and make goals."
Antoine Griezmann, Clement Lenglet, Miralem Pjanic and Frenkie de Jong were left on the bench for the first 60 minutes of the match but coach Ronald Koeman was still able to name a strong side. United States international Sergino Dest made his second start of preseason, with Pique, Sergi Roberto and Riqui Puig also involved from the off.
Girona's Cristhian Stuani forced a fine early save from Inaki Pena and young left-back Alejandro Balde produced a key block to stop the follow-up going in before Barca took over. Balde was fouled as he danced into the box at the other end and Pique, taking the ball off Manaj, snuck the penalty in. Puig and Nils Mortimer then combined to set up Manaj's fourth goal of preseason.
With Manaj off the field due to his head injury, Girona pulled one back. Saiz chipped a penalty down the middle after Nils had handled the ball at a corner and the Girona forward almost equalised moments later, hitting the bar from 25 yards.
With Memphis on, though, there was a change in the atmosphere and Barca started to create chances with ease.
Memphis twice came close, drawing a save from Juan Carlos before driving just wide. Griezmann almost got in on the act, too, while Pjanic hammered the bar with a free-kick from 30 yards.
Griezmann was then brought down in the area and, as the crowd chanted 'Memphis,' the new signing took the ball to crown a memorable debut.