Rook endgame: how to win with extra passed pawn without a rook from the rear?

Today I came into the following position: lichess.org/Uahfg4v3/white#66
It looks like a classical position from a textbook, but to be honest, I don't know even what the true evaluation is. It seems for me that white must be winning, but my attempt to win proved doubtful. In this position

engine recommends a7, but after ...h5 is seems to me that the position is an ironclad draw, and the engine is also clueless. My own attempt to sharpen the position is also unconvincing: it certainly poses some practical problems, but lead to a draw if black plays perfectly (e.g. 46... Ke5 instead of 46... h5?). My question is: what are classical games with the position like on the board and how to play them for a win?

This is a classical one with +1 pawn in a rook endgame and two very good players decided it was a draw :-)

lichess.org/broadcast/2018-chess-world-championship/fCum15PU/x1qOQ1iK

This is not the same kind of position. Rook endgames with one extra pawn on one side are known to be generally drawn, but when the extra pawn is far away on another side, it gets tricky.

FFS, I've lost my post twice because the tab refreshed.

tl;dr 41. f3 may have been winning, here's a study with some of the lines (follow with Stockfish and turn on infinite analysis; you'll hopefully find most of my analysis in the cloud if you select 2 variations): lichess.org/study/L6stQP9T

Of course, this was only made possible by the way black mispositioned his pawns (neither solid nor aggressive - the automated analysis didn't notice, but 39...g6 was the move that made his position precarious) and king (there was no reason not to play 38...Kf5, which would have enabled him to kick your king out of f3 with Ra3+).

I think 40. Ke3 was drawn against perfect play, but winning if black commits one or two inaccuracies.

The textbook game you were thinking of: http://www.chessgames.com/perl/chessgame?gid=1012518

The white King goes to the Queen‘s wing, giving up his own wing and it depends. As far as I remember such an endgame is analyzed in Dvoretzky‘s manual over 12 pages...

@Velysine I'll take a look on the analysis a bit later (and also Dvoretzky), but I don't actually think Alekhine - Capablanca is relevant here. I would be glad to take advantage of the Tarrasch rule and place my rook behind the pawn, but I had no opportunity to do this. And this completely changes the character of the struggle.

The white king must run to a7, then help promote so that black must give up his rook for the pawn, then rook and king must return in time to stop the black pawns. That is the general idea, but white must be careful that he can complete the last part.