Substitute Nicolas Pepe scored two outstanding free-kicks in the last 10 minutes of the game to rescue Arsenal as they came from behind to beat Vitoria Guimaraes at the Emirates Stadium.
Former Tottenham midfielder Marcus Edwards showed excellent composure to give the Portuguese side a shock early lead, before Brazilian teenager Gabriel Martinelli drew Arsenal level after half an hour.
The visitors regained the lead just four minutes later after Davidson's left-footed shot came back off the post and Bruno Duarte smashed in the rebound.
Substitute Nicolas Pepe equalised late on for the Gunners with a 25-yard free-kick before curling another into the top corner seconds before the full-time whistle.
Arsenal made it seven wins from seven games for English sides in Europe this week and with it remain top of Group F on nine points.
Unai Emery made 10 changes with only Joe Willock surviving from Monday's 1-0 Premier League defeat to Sheffield United while Alexandre Lacazette made his first start since 1 September.
Former Tottenham and England Under-20 international Marcus Edwards gave Vitoria Guimaraes a surprise lead in the 8th minute. Victor Garcia cut-back, before Edwards, who left Spurs in the summer, pulled it back onto his right foot and stroked it past Emiliano Martinez for his first goal for the club.
Arsenal were had an immediate chance to restore parity but Lacazatte dragged his shot wide from just outside the box.
Gabriel Martinelli pulled Arsenal level in the 32nd minute. Kieran Tierney put in a fantastic cross from the left and the 18-year-old headed the ball over Miguel Silva, who was well off his line, and into the empty net.
Within seconds Willock almost put Arsenal ahead but his shot was just wide of the post.
Five minutes later and Vitoria were back in front through Bruno Duarte. Arsenal gave the ball away in midfield and Davidson flew up the pitch before unleashing a brilliant shot from the edge of the area which cannoned back off the post. Before Martinez could pick himself off the ground, Duarte blasted in the rebound from a tight angle.
Arsenal again had an immediate chance to make matters level as Emile Smith Rowe picked out Martinelli, but he dragged his shot just wide of the far post.
Davidson almost made it 3-1 just before the break but his sweet strike flew just wide from 20 yards out.
The second half began with Vitoria having another chance to score a third. Davidson picked out Denis Poha, whose flicked shot was saved by Martinez.
After a much quieter second half, substitute Nicolas Pepe made it 2-2 with ten minutes remaining with a beauty of a free-kick. From 25-yards out, the wall and goalkeeper were expecting him to shoot at the near post, but Pepe watched keeper Miguel Silva take a step to his left, and then curled it across goal into the far corner.
Rob Holding had a chance to put the home side in front five minutes later from a delightful Pepe free-kick. He picked out the defender at the back post but his header on goal was gathered by Silva.
Pepe did it again in added time to give Arsenal a 3-2 win, curling a brilliant free-kick into the top corner.