I'm the dealer. The person to the left tells me to pick it up. He plays before I discard. As a result of the fast play, I forget to discard, and in the end of the play, I'm obviously holding onto one extra card. Does the official play begin at the discard? Or am I penalized for not discarding?
Officially play does not begin until the dealer has discarded.
(take the following quote with a grain of salt, there are tons of euchre variants and home rules that may contradict this).
XIII. If the eldest-hand leads before the dealer has discarded, he cannot withdraw his card and change his lead, nor can the dealer be deprived of his right to play alone. The discard is not completed until the dealer places his card under the kitty, or on the table, and has quitted it; and when the dealer has once quitted the discarded card, he cannot change it.
XVIII. If any player plays with six or more cards, or, if the dealer plays and omits to discard and fails to announce the fact before three tricks have been turned, such player or dealer cannot count the point, or points, made on their side in the hand, and they lose the deal. But if the adverse party win under such circumstances, they are entitled to count all they make.
If you had stopped play and discarded after the first card(s) were played, they cannot be un-play their card once you discard (and you can use that information to choose a discard).
Since you did not discard and played the hand with 6 cards, you cannot score points this hand.