How could I have won this Q vs R+N endgame?

Question

I was playing as White in this superior position when I decided to sacrifice my Queen for Knight + Pawn:

[FEN "8/2k1r1p1/5n2/5PK1/6PP/3Q4/8/8 w - - 0 46"]

 1. Qc3+ Kd8           
 2. Qxf6 gxf6+         
 3. Kxf6 Ke8           
 4. g5 Rf7+          
 5. Kg6 Ke7           
 6. f6+ Kf8           
 7. h5 Ra7           
 8. h6 Kg8           
 9. Kf5 Kh7           
 10. g6+ Kxh6          
 11. f7 Kg7           
 12. Ke6 Ra6+          
 13. Ke7 Ra8 0-1

My thoughts were that the 3 far advanced, connected pawns would beat the rook, but I was wrong.

My question is: what is the best move and plan in the starting position shown?

Answer 1

Partial answer - a Nalimov tablebase says your queen sacrifice should have resulted in a draw. Black cannot withstand your pawnroller and goes for a perpetual/stalemate combo. Your fatal mistake was 9. Kf5

[FEN "6k1/r7/5PKP/6P1/8/8/8/8 w - - 0 9"]

1.Kh5 (1.Kf5??) Kh7 2.g6 Kh8 3.f7 Ra5+ 4.Kh4 Rh5+

From the starting position, Stockfish recommended...

[FEN "8/2k1r1p1/5n2/5PK1/6PP/3Q4/8/8 w - - 0 46"]

1.Kg6 Nd7 (1... Nxg4?? 2.Qc4+ {Black can't hold the g-pawn and both white pawns will queen... it's a merciful mate in 17}) 2.Qc3+ Kd8 3.Qa5+ Kc8 4.g5 Ne5+ 5.Kh5 Nd7 {This shuffling moves shows that Black is out of ideas.} 6.f6 gxf6 7.Qc3+ Kd8 8.gxf6 Re4 9.Kg5 Re6 10.Kg6 Ra6 {Black could take the pawn, trading both his pieces for the Queen and a pawn. But then White Queens the h pawn. } 11.h5 Ra4 12.Kg5 Ra2 13.h6 Rg2+ 14.Kf5 Rh2 15.Kg6 Rg2+ 16.Kf7 Rh2 17.Qa5+ Kc8 18.Ke7 Nb8 19.Qf5+ Kb7 20.Qb5+ Ka7 21.f7 Na6 22.f8=Q Rc2 23.Qf6 Rc7+

Yep, that's pretty computer-tactical. But what you see is White pushing the King away from the pawns, and then relentlessly pushing the pawns. As an aside, it would take quite a player to separate their King from the Rook without losing the Rook to a queen tactic. Not recommended!

Here's what Stockfish thinks about the sacrifice.

[FEN "8/2k1r1p1/5n2/5PK1/6PP/3Q4/8/8 w - - 0 46"]

1.Qc3+ Kd8 2.Qxf6 gxf6+ 3.Kxf6 Re4 4.h5 Rxg4 5.h6 Rg1 6.h7 Rh1 7.Kg7 Rg1+ 8.Kf6 {Draw by repetition.} )