The Merseyside-born amateurs put their club on the map by winning the Claret Jug twice in three years, with Ball's victory in 1890 followed by Hilton's triumph two years later.
Their remarkable success in golf's original major exemplified Royal Liverpool's proud amateur history as the future Championship venue became woven into the fabric of The Open.
Established in 1869, Royal Liverpool's pedigree for elite non-professional golf stretches back to 1885 when it hosted the inaugural men's amateur championship - which then became The Amateur Championship.
Five years on, Ball won his second Amateur Championship on his home course before becoming the first Englishman and amateur to win The Open at Prestwick.
Hilton, who was born a mile from the Royal Liverpool clubhouse in West Kirby, went on to be crowned Champion Golfer himself in 1892 - the first time The Open was contested over 72 holes.
The decision to extend the Championship, which had previously been played over 36 holes, appears to have followed a flurry of articles and letters that appeared after The Open of 1891.
The two-day affair was staged at the new home of the Honourable Company of Edinburgh Golfers, 20 miles east of Musselburgh Links, where they had previously hosted six Opens.
Hilton had only opted to play the Championship at the eleventh hour, travelling up the day before after persuading his father to grant him leave from the family business.
And it was actually Ball who started the better of the two Royal Liverpool amateurs, firing rounds of 75 and 80 on the opening day to trail fellow amateur Horace Hutchinson by three shots.
Hilton found himself seven strokes behind the leader after rounds of 78 and 81, with most of the damage done with an outward 45 in his second 18 after lunch.
Sandy Herd and two-time Champion Golfer Willie Park Jr were the leading professionals going into the second day of action, sitting in a tie with Ball for second place with 155.
Hutchinson faded away on day two, dropping down the leaderboard with rounds of 86 and 80, as Ball usurped him at the top of the standings by posting 74 in his third round.
Hilton also surged into the title mix with the best round of the second morning, posting a 72 to sit just two shots behind his fellow club member going into the final 18 holes.
Ball closed with 79 to finish in a share of second place with Herd and defending Champion Golfer Hugh Kirkaldy, both players closing out with sixes on the last for a total of 308.
As the last of the leading players to start, Hilton started slowly with a six on the second but held his nerve to finish with a round of 74 - even with a six on the final hole - and win by three strokes.