According to The Oxford Companion To Food (1999 ed.):
Originally, in France, a sauté was a dish of meat of poultry cut
into pieces and cooked only in fat, but the French now also use the
term for dishes which simply involve browning a food before adding a
It also says:
The word ... has succeeded in migrating to English in both [noun and verb], with
the same accent.
So, the answer is that it did come from French, because it was a word in French for a cooking technique, and crossed over to English for the same technique.
The Oxford doesn't give a year for when this crossover supposedly occurred, but my guess is sometime in the mid-19th century. This is because the word originated in French supposedly in the early 1800's, so it couldn't be older than that. And the English version is likely to be younger than the popularity of the dish
Sauté d’agneau, because that dish uses the later French meaning of the word, to fry and then cook in liquid, which is "braising" in English and definitely not sauteing.
But we can narrow it down further. The first mention of the word I can find in an English-language cookbook is in the 1827 edition of The Art of French Cookery. It does not appear in the 1822 edition of The Cook's Oracle, nor in the 1823 edition of Domestic Cookery. So it seems to have appeared as a French term in British & American recipes the 1820's and been appropriated as an English term sometime later.
ADDED TO ANSWER:
There's good reason to believe that the sauté technique, at least as a widespread practice, did not exist for more than a few decades earlier than the word in French did.
The reason for this has to do with cooking technology. Through most of the 18th century, a majority of cooking was still done on open hearths, using a spider pan rather than a frying pan on a stove. Moving the pan to make ingredients "jump" -- an essential element of sauteing -- is very difficult with a spider.
So sauté as a common cooking technique would not have preceded iron stoves as common cooking surfaces, which didn't happen until the end of the 18th Century.