Arsenal had taken the lead against Sporting through Granit Xhaka, only to see it cancelled out by Pedro Goncalves' stunning 46-yard chip.
Eventually, Gabriel Martinelli's spot-kick was saved in the shootout, ending a run of six games undefeated in all competitions for the Gunners.
"We didn't win enough duels in certain spaces. We didn't put the ball under pressure well enough. We gave the ball away consistently and it created a really open game we didn't want to play," Arteta said.