Gabriel Martinelli's spot-kick was saved by goalkeeper Adan before Nuno Santos netted the winner to spark celebrations among Sporting's players following an eventful and entertaining tie.
A sensational long-range chip by Pedro Goncalves from 46 yards forced extra time after Arsenal's Granit Xhaka had earlier pounced on a rebound for the opener.
Arsenal defender Gabriel came agonisingly close twice in the final minutes of extra time when his header was tipped over the bar, and then he had an effort cleared off the goalline.
A sensational long-range chip by Pedro Goncalves from 46 yards forced extra time after Arsenal's Granit Xhaka had earlier pounced on a rebound for the opener.
Arsenal defender Gabriel came agonisingly close twice in the final minutes of extra time when his header was tipped over the bar, and then he had an effort cleared off the goalline.