Broady and Farquharson are the first British winners of the title since James Trotman and Martin Lee in 1995.
Burton was broken again in the third game of the second set and that proved enough for Broady and Farquharson.
Once Broady and Farquharson had picked up their early break in the second set, Burton and Morgan were chasing the match.
They did have two break points on Broady's serve in the sixth game but he saved them both - the second with a 93mph ace.
Britain's Liam Broady and Tom Farquharson also enjoyed a comeback win as they overcame United States third seeds Denis Kudla and Raymond Sarmiento 4-6 6-2 6-2.
BBC: GB's Golding & Robson into Wimbledon junior semi-finals
Broady is from Stockport, hometown of Britain's last Wimbledon men's singles champion Fred Perry, and said there could easily have been more British junior success at SW19 this year.
Broady was broken in the opening game of a tight first set but Burton was broken back in the sixth game and it was finally decided in a tie-break.
The winners were Oliver Golding, James Marsalek, Laura Robson, Tara Moore and Eleanor Dean, while Liam Broady, Lewis Burton, Ashley Hewitt and Fran Stephenson all produced encouraging displays against top-10 ranked opponents.
Despite lacking consistency at present, 16-year-old Broady - a recent practice partner for Rafael Nadal - possesses an impressive physique, has a calm on-court demeanour, is light on his feet and benefits from a potent forehand.
Broady and Farquharson had three breakback points in Burton's next service game and took the second to level the set at 3-3 and Morgan saved two break points in his next service game with two big serves.
Farquharson saved an early break point in the opening game with a smash by the net and although Broady had an advantage he lost it and then served a double fault to give his opponents the early initiative.
And while Hewitt, Burton and Broady all fell to defeats at Wimbledon that did not reflect the quality of their play, Burton wowed the Court 14 crowd with serve-volley tactics and Broady illustrated the all-round promise that ought to take him far.
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