The Nicki Minaj/A.1. Steak Sauce love affair isn't a one way thing ... because it turns out the folks behind the tasty, tasty sauce are as big a fan of her as she is of them.

If you'll recall ... our photogs spotted Minaj out this week at Boa -- a super-fancy steakhouse in Hollywood -- where she had someone bring her a bottle of A.1. into the restaurant.

While the move is a HUGE no-no in steak etiquette, the folks over A.1. tell TMZ they were thrilled to see Minaj make a sauce run and they are HUGE fans of hers.

The rep tells us, "We’re so glad to hear that Nicki Minaj loves our sauce just as much as we love her and her music. To make sure she’s never 'sauceless' again, we sent Nicki some of our A.1. Original Steak Sauce. Now she’ll have A.1. wherever she goes."

Much to delight of Nicki ... and the chagrin of chefs everywhere.