The term “top dog” means to be the boss (or leader), or to be victorious. The term “top dog in the fight” was cited frequently in the early 1860s.
   
“Whether he’s top dog or bottom dog/ Or spectator dog in the fight” was printed in The Louisiana Democrat (Alexandria, LA) on July 13, 1859. “The top dog in the fight” was printed in the Petaluma (CA) Argus on July 28, 1863.
 
“Top dog” possibly comes from dog fighting and is associated with the term “under dog.”
 
   
Wikipedia: Top Dog
The expression top dog is an idiom for the boss or the leader. In a competition, it is also the favorite or the one expected to win, and the opposite of the underdog. It may be a shorthand reference for a dominance hierarchy.
     
Wikipedia: Dog fighting
Dog fighting is a type of blood sport generally defined as two or more game dogs against one another in a ring or a pit for the entertainment of the spectators or the gratification of the dogfighters, who are sometimes referred to as dogmen.
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In 1817, the Staffordshire Bull Terrier dog breed was brought to America and dog fighting slowly became part of American culture. Yet, though historical accounts of dog fighting in America can be dated back to the 1750s, it was not until the end of the Civil War (1861–1865) that widespread interest and participation in the blood sport began in the United States.
 
(Oxford English Dictionary)
top dog  n. literal the dog uppermost or ‘on top’ in a fight; figurative the victorious or dominant party.
1900   Speaker 28 Apr. 97/1   The most popular argument in favour of the war is that it will make the individual Briton top dog in South Africa.
1906   P. White Eight Guests (Tauchn.) I. 66   Marcus had never had a tussle yet without coming out ‘top dog!’
1906   Daily Chron. 26 Mar. 6/4   I recall..many in which I started as under-dog and came out top-dog.
     
Chronicling America
13 July 1859, The Louisiana Democrat (Alexandria, LA), “Dawgs,” pg. 1, col. 3:
If a “feller” must be a dog,
It matters never a mite
Whether he’s top dog or bottom dog
Or spectator dog in the fight.
     
28 July 1863, Petaluma (CA) Argus, “Tog Heavy Bait,” pg. 1, col. 5:
The top dog in the fight was small, but he was all yellow. (...)—Sierra, Democrat.
 
2 July 1864, Boston (MA) Herald, “Police Court—Before Judge Wright,” pg. 4, col. 3:
But this morning Billy was the “top dog in the fight,” for Tim was fined $25 and costs.
 
Chronicling America
12 November 1864, Ashtabula (Ohio) Weekly Telegraph, pg. 3, col. 3:
Deputy Marshal Dickinson in an effort to stop the melee caught one of our hands, and held it while waiting for some one to remove the top dog in the fight.
     
24 November 1865, The Daily Examiner (San Francisco, CA), “Choked Off,” pg. 2, col. 2:
We fear, however, the Bulletin is the top dog in the fight.
   
Google Books
19 May 1866, Wilkes’ Spirit of the Times (New York, NY), pg. 180, col. 3:
It was now apparent to all present that Collier would prove the top-dog in the fight.
 
Google Books
Modern Thinkers Principally Upon Social Science:
What They Think, and Why
By Van Buren Denslow
Chicago, IL: Belford, Clarke & Co.
1880
Pg. 248:
But if the under dog in the social fight runs away with a bone in violation of superior force, the top dog runs after him bellowing, “Thou shalt not steal,” and all the other top dogs unite in bellowing, “This is divine law and not dog law;” the verdict of the top dog so far as law, religion, and other forms of brute force are concerned, settles the question.
 
OCLC WorldCat record
The top-dog.
Author: Fergus Hume
Publisher: London : [F.V. White?], 1910.
Edition/Format:   Print book : Fiction : English
 
OCLC WorldCat record
Top Dog. One Step, etc. [P.F.].
Author: Corelli Windeatt
Publisher: London : Francis, Day & Hunter, 1911.
Edition/Format:   Musical score
 
OCLC WorldCat record
Top dog : and other sketches for women’s institutes, girl guides, etc., etc.
Publisher: London ; New York : S. French, ©1926.
Series: French’s acting edition, no. 92.
Edition/Format:   Print book : Fiction : English
 
OCLC WorldCat record
Top dog; the story of your economic future and welfare under the Taft-Hartley law and its program against labor.
Author: John C Stevenson
Publisher: [Los Angeles], [Parker & Son], [1955]
Edition/Format:   Print book : English
   
OCLC WorldCat record
Top dog : and other doggone delightful expressions
Author: Carli Davidson
Publisher: San Francisco, California : Chronicle Books, [2016]
Edition/Format:   Print book : Juvenile audience : English
Summary:
From the lens of expert animal photographer and New York Times bestselling author Carli Davidson, an adorable cast of canine characters star in this doggone delightful tribute to everyone’s favorite idioms, from “working like a dog” to “the dog days of summer.” These endearing and hilarious images are sure to make any animal enthusiast smile in “two shakes of a dog’s tail!”
     
27 January 2018, The Wall Street Journal, “Word on the Street: Underdogs Beyond the Super Bowl” by Ben Zimmer, pg. C4:
The poem, entitled “The Under Dog in the Fight,” was first published on April 4, 1859, in the New York Evening Post and later republished in Washington, D.C. and other places. The poem was sent in by David Barker, a lawyer from Exeter, Maine, who specialized in homespun verse. His poem includes the lines, “But for me—and I care not a single fig / If they say I am wrong or right—/ I shall always go for the weaker dog / For the under dog in the fight.”
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As Barker’s original verse became forgotten, other explanations for the origins of “underdog” have sprung up over the years. One popular theory, seen in some museum displays, relates the term to the old practice of sawing wooden planks in a saw pit, with men working the ends of a long two-handed saw. The senior man at the saw’s top handle was known as the “top dog,” while the junior man, stuck in the pit with the sawdust, was the “under dog.”
 
But while saw pits really did work that way, there is no evidence that “underdog” came out of that context. The 1859 poem is still the scrappy winner of this etymological fight.