The situation is really quite simple to resolve. It's called a "replay". What you would do is have the teams come back the next day, or the day after, and have another possible 120 minutes for the teams to settle things.
Hear me out. One of the practical reasons for the abolition of replays at the World Cup in particular was the logistics involved in them; but I think we've moved on a little since 1970, no? That's just a minor point, though. Here's the killer. From 1930 to 1962 (the period of time when a draw after extra time at the World Cup resulted in a replay; in 1966 it was replaced with drawing of lots except the final), there were 50 knockout matches played, of which precisely three required a replay, one in 1934 and two in 1938. Italy weren't affected after their replay in 1934 (for reasons which everybody knows), but in 1938, both teams who won replays were knocked out in the next round.
And that's why replays are best; they're like a nuclear war, in that neither team comes out of them in good shape. Sure, somebody eventually wins and advances, but that then means that they've played at least 90 minutes' more football than their opponents, and that fatigue is going to catch up with them very quickly. Therefore it's to neither team's benefit to just play for a draw, and they had to come out and attack at some point, so extra time was very rarely dull because of that looming threat of a replay.
You know it makes sense...