Dwight Gayle's 20th goal of the season put Rafa Benitez's side ahead in the first half at Griffin Park.
Lasse Vibe's equaliser after the break pegged them back, but substitute Murphy nodded home the winner 11 minutes from time to give his side a precious victory in the capital.
Benitez made eight changes to the side that drew 1-1 at Birmingham City in the FA Cup last time out, with only captain Jamaal Lascelles, full back DeAndre Yedlin and midfielder Jack Colback keeping their places.
And it took just seconds for United to impose themselves, as Yedlin played in leading scorer Gayle, whose deflected shot nicked the sidenetting.
Successive corners then saw the visitors apply more pressure with both Lascelles and Matt Ritchie threatening to convert some dangerous crosses, and on the 20-minute mark they made a deserved breakthrough.
Gayle held his line well to stay onside, picking up the ball wide on the left. The former Crystal Palace man then powered into the box before unleashing an unstoppable left-footed drive to beat goalkeeper Daniel Bentley from a tight angle in front of over 1,600 travelling Magpies supporters behind that goal.
But Brentford rallied, and Vibe called Karl Darlow into action with a low effort, with the ex-Nottingham Forest man rushing out to block the shot.
Just eight minutes after his opener, Gayle was forced off through injury with Murphy - a scorer against Birmingham last weekend - sent on in his place just before Ayoze Perez was denied a second by a last-ditch Bentley save.
The Bees continued to press, though, and Darlow had to save twice more from Vibe - firstly repelling a venemous shot when the forward went one-on-one, before coming off his line once more to narrow the angle with Vibe poised to shoot, smothering his eventual strike.
And just before the break, Nico Yennaris drove just past the upright from distance as the hosts ended the half in the ascendency.
And Dean Smith's side began the second half as they had finished the first. After the impressive Ryan Woods won a corner, Darlow was forced to make an instinctive save from close range, but Vibe managed to scramble the ball home to equalise just eight minutes into the second period.
Vibe then saw his cross-shot cut out by Ciaran Clark before glancing the resulting corner from Woods just past the post as Benitez's side struggled to find their attacking rhythm.
Vibe went so close to putting his side ahead just after the hour mark, when he latched on to Romaine Sawers' flick and beat Darlow, but his shot hit the upright, rolled along the goalline and Lascelles bravely hacked it clear under pressure from Yennaris.
Isaac Hayden hobbled off and was replaced by Vurnon Anita shortly after, but then Anita himself picked up an injury after a challenge from John Egan and was stretchered off, with Grant Hanley coming on in his place.
It was another blow for the Magpies, who had to be alert to block another Vibe effort with just under 20 minutes remaining before the Danish frontman stabbed just wide of the target after turning well in the box.
But they remained resolute, and when Perez swung in an enticing cross with the outside of his right boot with just over ten minutes left of the clock, substitute Murphy got there before goalkeeper Bentley to nod in his second goal in as many games to hand United a vital three points.
