Before Galatasaray it was hit and miss. We went out to Leicester in the cup that same season thanks to a penalty shootout and we beat West Ham (if I'm not mistaken) in the run up to winning the cup in 98 after a penalty shootout.
The greatest one I've witnessed is the one against Sampdoria in 1995. They scored 2 late goals in extra time. We're doomed and out. Arsenal win a free kick out of nowhere. Stefan Schwarz, doing about the only thing he's ever done in an Arsenal shirt, buries the free kick (last kick of the game) making it 3-2 Sampdoria on the night, 5-5 on aggregate, and we win the ensuing shootout.
There was an incredibly interesting stat I read that for any English team in Europe, or indeed the England team in a penalty shootout, if they were shooting on the left hand side of the TV screen they'd always lose, and if it was the right, they'd win. This was consistent for a good 15-20 years