Sometimes violating opening principles is one's best way of playing for an advantage. Even if you are not the one playing for advantage, opponent might be.
The first opening that comes to mind is the Two Knights Defense:
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]
1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 {Black's played completely according to principles so far - "develop knights before bishops", etc.} 4. Ng5!? {This is White's best way of playing for an advantage, but it violates the opening principle of not moving the same piece twice.} d5 {It's either this or 4...Bc5 giving up the f7-pawn, which is an even riskier move.} 5. exd5 Nxd5?! {This move is known to be dubious, but the main line 5...Na5 is effectively a pawn sacrifice.} 6. d4
With a powerful initiative. White doesn't have a forced win, but has open lines and is ready to castle, with threats such as Qf3 hitting both the d5-knight and f7-pawn.
A different line of the Giuoco Piano comes to mind as well.
[FEN ""]
1. e4 e5
2. Nf3 Nc6
3. Bc4 Bc5
4. c3 Nf6
5. d4 exd4
6. e5
Black's best move is 6...d5 (the alternatives are nearly losing), which is not an obvious move and not something that you can find knowing only opening principles.