Leeds were knocked off the top of the Championship table following a 2-0 defeat at home to Sheffield Wednesday.
But they found the second half tougher going as Wednesday successfully stifled their creativity.
And the South Yorkshire side were given their rewards when Murphy beat Kiko Casilla from a narrow angle.
Nuhiu wrapped the points up four minutes into stoppage time after Adam Reach teed him up.
It left Leeds six points ahead of third-placed Brentford in the race for automatic promotion.
Bamford failed to get hold of his shot after he had won the ball back, while Murphy fired a first-time shot wide after Liam Cooper did enough to put him off.
Harrison volleyed wide from close range when Kalvin Phillips picked him out.
Bamford had a header chalked off five minutes later after Mateusz Klich was ruled offside in the build-up.
Dawson saved a Bamford shot at his near post and Costa smashed over in stoppage time.
Bamford shot over three minutes after the restart after Costa got to the byline and picked him out.
Costa failed to find the target as Leeds were once again restricted to shots from long range.
Dawson saved a curling effort from Klich four minutes later, before Casilla had to turn away Murphy's goalbound inswinging corner.
Reach fired over with a cross-shot in the 74th minute, before Harrison missed the target with a header with 10 minutes to go.
Murphy's low shot squirmed past Casilla in the 87th minute to give Wednesday the lead, and it was game over when Reach got in behind the defence and set up Nuhiu deep into stoppage time.