Cribbage score going out
It's fine to go over in standard cribbage; the game is won by the first player or partnership to reach or exceed 121 points.

But remember the way things are scored: as they happen.

If that three points you mention was in play, such as opponent leads a 9, you play a 5, opponent plays a 6, and you play a 7, scoring three for the run in play, then the go from the opponent puts another 1 point score on the pile.

If it's a run in hand, however, it's scored only after the last card, starting with the player to the dealer's left and going left around the table (same as play order). If you are at 117 points, have a 3 point hand (say AJQK and a 2 up), and get the last card for 118, and the other guy is at 110 and gets 16 (AA23), it is determined by who scores hand first. If you dealt, you lose; if he dealt, you win. In three or four player non-partnership, if he's to your right, you've lost.

And remember: muggins can't be claimed until after the peg has been moved. Only on an overpeg would it prevent victory.

Also remember: if your scoring sheet shows last card separate from Go, there is no "go" on last card.

There is a common variant, called exact out, where you have to hit 121 exactly, and any score that would take one over is lost (but must be called out to prevent it being muggins-claimed).