Arsenal crashed out of the Europa League after a penalty shoot-out defeat to Sporting CP in the Europa League round-of-16 at Emirates Stadium.
Granit Xhaka handed the Gunners a first-half lead but that advantage was cancelled out in spectacular style by Pote's long-range effort that sent the tie into extra time.
Adan ensured the tie went the full distance with crucial saves to deny Leandro Trossard and Gabriel Magalhaes, though Sporting finished with 10 men after Manuel Ugarte's dismissal.
Adan's heroics came again in the showdown from the spot, denying Gabriel Martinelli's attempt before Nuno Santos scored to send Sporting into the quarter-finals.
The Gunners took the lead after 19 minutes, with Xhaka playing a superb ball over the top for Martinelli to force Adan into a save but the Switzerland international was on hand to smash home the rebound.
Sporting started the second period on the front foot and found the equaliser in stunning fashion shortly after the hour mark, Pote spotting Ramsdale off his line and lobbing the England international from 46 yards out.
The Gunners stopper redeemed himself 10 minutes later, charging off his line to block Marcus Edwards' effort with his face after the former Tottenham youngster was fed through by Ricardo Esgaio.
Trossard was gifted a golden opportunity in extra time when Youssef Chermiti sloppily gave the ball away but Adan denied the Belgian and deflected his effort onto the post.
The Sporting stopper was again on hand to deny Gabriel's powerful header, the Brazilian then seeing another attempt from the following corner cleared off the line before Ugarte saw red after a second booking due to clattering into Bukayo Saka as the game headed to penalties.
With the first seven penalties of the shoot-out scored, Adan thwarted Arsenal again, diving low to his left to deny Martinelli before Nuno Santos comfortably dispatched his attempt to seal a spot in the last-eight.