No one knows when the first sauce was applied to barbecued meats. The first advertisement for commercially made barbecue sauce appears to be for Georgia Barbecue Sauce in 1909.
 
“A1 International Sauce”—now popularly known as “steak sauce”—dates from 1862, with the name “A1 Steak Sauce” applied in 1873.
     
   
Wikipedia: Barbecue Sauce
The U.S. has a wide variety of differing barbecue sauce tastes:

Kansas City – thick, red-brown, tomato, molasses
North Carolina – liquidy vinegar, pepper flakes
South Carolina – mustard, vinegar, black pepper, light or thick tomato
Alabama – traditionally mustard and vinegar based and seasoned with roasted or smoked chile peppers, although a white, mayonnaise based sauce is equally popular in the Northwest regions of the state
Georgia – a tremendous variety exists within the state, but “traditional” Georgia barbecue sauce features a ketchup base flavored with garlic, onion, black pepper, brown sugar, and occasionally bourbon
Arkansas – thin vinegar and tomato base, spiced with pepper and slightly sweetened by molasses
Texas – tomato based with hot chiles, cumin, less sweet
St. Louis – generally tomato-based, thinned with vinegar, sweet and spicy. It is not as sweet and thick as Kansas City-style barbecue sauce, nor as spicy-hot and thin as Texas-style.
Wisconsin – smoky taste, spicy, and not sweet.
 
Wikipedia: Steak Sauce
Steak sauce is a generic term used in the United States for various premade meat sauces. Most commonly, it refers to sauces served with beef and of western origin. The term “steak sauce” is derived from these sauces often being served with steak in restaurants.

The most prominent brand of steak sauce in the US is A1 Steak Sauce. In the United Kingdom and Canada, HP Sauce is more widely sold. In the UK, “brown sauce” is the generic term for HP-type sauces.

These sauces are normally brown or orange in color and often made from tomatoes, spices, vinegar, and raisins. Some contain anchovies. The taste is either tart or sweet with a peppery taste similar to Worcestershire sauce, made by Lea & Perrins.
 
A1 Steak Sauce History
1824-1831 In a time when a kings palate was supreme. Henderson William Brand, chef to England’s King George IV, creates a special sauce for the king’s table. The king is so impressed with the new sauce, he proclaims it “A1”.
1831 H.W. Brand knowing that he had a great idea leaves the palace walls and starts Brand & Co. where he begins to manufacture meat extracts and essences.
1850 Being a great culinary master, but not a businessman, Brand goes bankrupt and has to sell his business to his friend, W.H. Withall. Knowing that Brands products are the best, Withall continues to do business under the Brand & Co. name.
1862 Steak Sauce is entered at the International Exposition in London, England. The sauce is again proclaimed “A1” . The sauce with its royal beginnings is on its way to be the most famous steak sauce in North America.
1873 Withall decides to sell Brand & Co. to Dence and Mason. Brand re-enters the picture claiming trademark infringements. Legal battles will continue for another eight years until Dence and Mason agree to place Brand on an annuity and purchase Brand & Co. The steak sauce is now officially called A1 Steak Sauce®.
   
29 December 1907, Atlanta (GA) Constitution, “Wilkes Convicts Eat Barbecue Cooked by World’s Best Chef,” pg. 3:
Two carcasses were required to feed the gathering, which, together with the twenty-five loaves of bread, several gallons of hash and a like quantity of Brunswick stew and a liberal quantity of the barbecue sauce to give flavor to the meats, made up a menu calculated to suit the most exacting requirements of an epicure.
 
31 January 1909, Atlanta (GA) Constitution, pg. A2 ad:
Georgia Barbecue Sauce
is the finest dressing known to culinary science for Beef, Pork, Mutton, Fish, Oysters, and Game of every kind; roasted, fried or broiled. It is also unequaled for perfecting Brunswick Stew and as dressing for Vegetables.
(...)
GEORGIA BARBECUE SAUCE CO.
ATLANTA, GA.
 
1 April 1909, Atlanta (GA) Constitution, pg. 9:
Roast Pork with Barbecue Sauce.
 
27 March 1934, Washington Post, “Ground Steak Goes High Hat And Delicious,” pg. 14:
An informal party may be made an occasion well spent if one can enjoy hamburger barbecues, slipped between soft rolls and flavored with a little relish of barbecue sauce.
 
8 February 1936, Washington Post, “Beef Extract Held to Add Zest to Food; Makes Even Ordinary Dish Fit Palate of the Epicure,” pg. 11:
Steak Sauce.
2 tablespoons butter
1/2 tablespoon liquid beef extract
1/2 tablespoon chopped parsley
Soften the butter, add the beef extract and chopped parsley and allow to melt on broiled chops or steaks.
 
(Trademark)
Word Mark   THE A1 INTERNATIONAL SAUCE
Goods and Services IC 029. US 046. G & S: SAUCES [ OR RELISHES ] FOR USE
WITH FOOD. FIRST USE: 18620000. FIRST USE IN COMMERCE: 1862000
Mark Drawing Code   (3) DESIGN PLUS WORDS, LETTERS, AND/OR NUMBERS
Design Search Code 261101
Serial Number   70026824
Filing Date June 1, 1895
Registration Number 0026824
Registration Date   July 16, 1895
Owner   (REGISTRANT) BRAND & CO. DENCE, THOMAS AND MASON, JOHN JAMES, BOTH
CITIZENS OF ENGLAND FIRM ENGLAND 11 LITTLE STANHOPE ST., LONDON ENGLAND(LAST
LISTED OWNER) DEL MONTE CORPORATION CORPORATION ASSIGNEE OF NEW YORK ONE
MARKET PLAZA SAN FRANCISCO CALIFORNIA 94105
Assignment Recorded ASSIGNMENT RECORDED
Type of Mark   TRADEMARK
Register   PRINCIPAL
Affidavit Text SECT 12C. SECT 15.
Renewal 4TH RENEWAL 19851015
Live/Dead Indicator LIVE