Newly-promoted Leeds looked to be cruising towards their first top-flight victory for 16 years when Bamford's precise finish put the Whites 3-1 up in the 50th minute of Saturday's clash at Elland Road before Helder Costa added United's fourth seven minutes later.
Costa had earlier put Marcelo Bielsa's side 1-0 up with a fierce strike in the fifth minute and while Aleksandar Mitrovic equalised from the penalty spot, a Mateusz Klich spot kick put United back in front.
But even after Costa's second goal had opened up a three-goal cushion, United then had to survive some nervy final moments after Fulham hit back with a strike from Bobby Decordova-Reid and header from Mitrovic which made the contest 4-3 with 23 minutes left.
Fulham's Neeskens Kebano then hit the post as the Whites and Cottagers continued to exchange attacks in a game resembling a basketball match but Leeds held firm to secure a historic victory, one week after being involved in another 4-3 thriller at defending champions Liverpool who saw off United at Anfield by the odd goal in seven.
Asked if it would be 4-3 every week, Bamford told BT Sport: "I hope not. I feel like last weekend was kind of like a free hit.
"We were playing against the best team in the country and we kind of showed a good side of ourselves, there were positives and negatives.
"But then this was obviously a crucial match against another team that came up last year and we wanted to win but that was a little bit too close in the end.
"I think that there's going to be a few long meetings this week just to pick out what we can do better but we won, that's the main thing."
Bamford was taken off in the 70th minute for Gjanni Alioski and the Whites no 9 admits he was not exactly enjoying seeing Leeds hang on for victory from the stands.
Reflecting on how United went from being 4-1 up to ultimately protecting a one-goal lead, Bamford reasoned: "I think that sometimes moments change the game and I think that second goal caught us off guard.
"The third one wasn't long after that so that kind of made us panic I think a little bit and obviously we managed in the end to see it out.
"But watching from the sidelines is horrible.
"It wasn't a nice watch when I came off."
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Thank you Laura Collins