The pair were made available for selection as they continue their comeback from injury by United head coach Marcelo Bielsa for the trip to Northumberland as Bamford came through 45 minutes as a half-time substitute with Brown completing 80 minutes of the fixture.
United dominated the ball early on as Brown worked goalkeeper Nathan Harker twice in quick succession in the early exchanges.
It was the hosts though who opened the scoring with their first opportunity of the evening.
A ball over the top found Thomas Allan who made his way into the box to drill a low cross across goal which fell to Elias Sorenson at the back post who tapped home with the net gaping.
Leeds responded to the setback and should've been level at the break as Leif Davis picked out Kun Mihaylov inside the area but the striker somehow mishit his volley with the goal at his mercy.
Carlos Corberan's men continued their good play after the break following the introduction of Bamford from the bench as he saw his first action since early September.
Midfielder Brown moved over to the left and caused the Magpies all sorts of issues cutting inside onto his right foot as the hosts failed to deal with his trickery.
Bamford himself saw a header blocked before defender Dean Burlace also had an effort swept away as Leeds pressed on in search of a leveller.
It was Newcastle again who had the ball in the back of the net in the dying moments as Matthew Longstaff flicked a shot over the stranded Kamil Miazek only to see the offside flag raised after much confusion from the officials with the hosts already celebrating.
They finally had their second goal though as the clocked ticked into injury time as substitute Adam Wilson burst through on goal with Leeds caught short at the back having pushed men on.
The striker calmly drilled the ball underneath the onrushing Miazek to seal all three points and hand the Whites their second defeat of the Premier League Cup to leave them with a mountain to climb in search of qualification from Group H.
Leeds United XI: Miazek, Struijk, Casey, Diaz, Davis, Burlace, Nicell, Brown, Oduor, Stevens, Temenuzhkov. Subs: Rae, Wollerton, Bamford, Dalby, Edmondson.
Newcastle United XI: Harker, Bailey, Gibson, Longstaff, O'Connor, Watts, Allan, Sangare, Sorensen, Cumbrera, Lazaar. Subs: Walters, Huuhtanen, Wilson, Longelo, Toure.