Hi Gal,
I’m sorry, I don’t think so.
Taking your summary of Marx as so, i.e. “Marx wrote that all things that have value was created by labour”, then Marx might say that, because Labor precedes, in time, the production of Valuable Things, that Labor Causes Valuable Things. To suppose the opposite (Value produces Labor) would seem backwards to Marx, since a Cause cannot precede its Effect.
Marx’s argument (as you present it) appears to be Post-Hoc ergo Propter Hoc at best. That is to say, that simply because two events precede each other in time, it does not logically imply a causal relation. E.G. Many singers take a drink of water before singing, therefore drinks cause singing.
This is not quite the Austrian complaint, however, since the Labor Theory of Value does not stop with the simple causal relationship you mention, but that the Amount of Value is a function of the Amount of Labor. E.G. A stack of wood is more valuable than a piece of wood because it requires MORE labor to produce a whole stack than 1 piece, or more commonly, it takes more labor to extract a diamond than to make a loaf of bread, therefore a diamond must be more valuable.
The Austrians adhere to the theory of subjective marginal utility, which is to say that the value that any one person puts on a given object is the subjective (personal) marginal (next use) utility (satisfaction of need) of the object to that person at that time. Prices are then a function of an aggregate bidding war between everyone who values given goods on the market. E.G. A single piece of bread being bid on by a full man and a starving man of equal means will likely be won by the starving man, since his personal next use value involves saving his very life, whereas the full man would derive only some momentary pleasure. The bakers own valuations on his creation is also a factor in this bidding, but you get the idea. Now imagine this situation going on on a massive scale, and you see where prices come from.
Now, I’m not sure the Austrians would agree simply that Value causes Labor, though I certainly think that sentiment is closer to the truth than the reverse. If I value a swim on a hot day, I will likely go through all manner of means of production to get my body into a cold river. 
In a more sophisticated case, an observant entrepreneur seeing the great value people place on video games might very well calculate that it is in his interest to procede to labor to produce more of them.
Hope this helps. Keep thinking and toying around with logical analysis as you are. It’s fun, and fruitful!