After Alexandre Lacazette was off target for Arsenal in the first half, Gosling punished slack defending when he turned in Jack Stacey’s cross from the right.
The Gunners could have been 2-0 down at the break against Eddie Howe’s spirited Bournemouth side.
Joshua King embarked on a strong solo run down the left and, after using his strength to hold off the last defender, almost managed to poke the ball through goalkeeper Bernd Leno’s legs from a narrow angle.
Arsenal improved after the interval and the Gagon striker Aubameyang finished first time after Reiss Nelson’s effort from the edge of the penalty area deflected into his path.
Substitute Joe Willock, who came on for the recalled Mesut Ozil, had a chance to win the game right at the end, but his injury-time shot was too close to the goalkeeper.
Arteta, a former Arsenal midfielder, was given a warm welcome from the traveling fans who are desperate to see signs of improvement.
He has much work to do after replacing Unai Emery and interim coach Freddie Ljunberg, with home games against Chelsea on Sunday and Manchester United on Jan. 1.
After learning the coaching ropes as assistant to Pep Guardiola at champion Manchester City, Arteta signed a 3 1/2-year contract. Emery was fired in late November after the team’s worst run in 27 years.
Arteta’s arrival could signal an unexpected boost for the erratic midfielder Ozil, whose form and perceived lack of work rate have frustrated Arsenal fans in recent seasons.
But Ozil has made a good first impression on his new coach, who was himself a fine passer, with both his attitude and creativity.
“To be fair his attitude in training since the day that I walked into the building has been incredible,’’ Arteta said. ‘’I said I was going to give a clean slate to everybody, and it was fair to give it to him as well, we could have put two or three goals into the net because of him.”