Whites head coach Marcelo Bielsa was clearly wary of Luton Town's threat and most creative influence Izzy Brown.
The former Whites loanee sits high in the Championship's assists charts and United's Kalvin Phillips was deployed to follow his every move.
With captain Liam Cooper back fit but only on the bench, a three-man back line of Ben White, Gaetano Berardi and Luke Ayling looked to keep out forwards James Collins and Harry
Cornick and the trio defended well though the Hatters still had chances in an open first half.
Ayling was required to produce a fine clearance to deny Collins from a Cornick cross and Sonny Bradley fired just wide from the corner.
The Kenilworth Road faithful also felt they should have had a penalty when Stuart Dallas sent Collins to the floor inside the box.
But after a quiet start, Leeds began to create chances at will and the half ended with a familiar theme of several Whites opportunities going begging.
Three stood out with a fine Gaetano Berardi through ball latched on to by Bamford whose flick beat keeper James Shea but not the best.
Leeds quickly recycled and Shea somehow twice kept out Harrison at the far post.
And from a wonderful move started by twinkle toes Harrison, Tyler Roberts was then thwarted by Shea who superbly tipped his effort around the post.
Leeds then immediately began the second half on the front foot and the Whites finally made the breakthrough in the 51st minute but not without furious scenes inside Kenilworth Road.
Brown went down outside the Whites box but referee John Brooks waved play on allowing Leeds to counter and Ben White played in Bamford who produced a brilliant finish into the bottom right corner.
But the lead lasted just three minutes as Luton's main man Brown produced a fine cross from the right from which James Collins headed home after escaping Ayling.
Luton then thought they had forced home a second moments later from a scramble inside the box only for a woblling Leeds to be saved by on offside flag.
United, though, looked to have weathered the storm and after Bamford and Hernandez failed to convert chances, Marcelo Bielsa brought on Jack Clarke on his 19th birthday as United's third and final change with Cooper and Helder Costa introduced too.
But it was United's no 9 Bamford who proved Leeds' saviour, just when United looked to have run out of time.
A dogged Luton looked to have ended a run of straight-losses but Leeds had other ides as Klich sent in a fine low cross from the right which Matt Pearson converted into his own net under pressure from Bamford to spark wild scenes in the away end.
Luton Town: Shea, Potts, Tunnicliffe, Bradley, Pearson, Shinnie (Bolton 89), Mpanzu, Bree, Brown (LuaLua 75), Cornick (McManaman 75), Collins. Subs not used: Sluga, Galloway, Jones, Butterfield.
Leeds United: Casilla, Ayling, White, Berardi (Cooper 59), Dallas, Phillips, Klich, Hernandez, Harrison (Costa 64), Roberts (Clarke 71), Bamford. Subs not used: Meslier, Douglas, Alioski, Gotts.Referee: John Brooks.
Attendance: 10,068 (1,035 Leeds).