After Dan Burn had a first-half goal controversially disallowed, Murphy's left-footed curler with ten minutes left looked to have snatched all three points for the Magpies at Craven Cottage.
But substitute Muniz's flicked finish just moments later pegged them back on an afternoon in which Matthias Jaissle's men turned in a better performance - but just couldn't quite find enough to hold on for the victory.
Head coach Jaissle made four changes to his United side for the trip to the capital, with Malick Thiaw, Sean Steur, Murphy and Yoane Wissa coming into the line-up.
Centre half Thiaw was called into action early on to hack away a dangerous cross from Oscar Bobb as Álvaro Arbeloa's charges threatened first, though Amar Dedić's willing running down the right was a useful outlet for the Magpies as they looked to return to winning ways after back-to-back defeats to Everton.
Lukáš Horníček was called into action for the first time to turn away Bobb's curling effort after 14 minutes, with Gonzalo García squandering a fine chance from the following corner as he lifted his header onto the roof of the net.
At the other end, Bernd Leno had to react swiftly repel Harvey Barnes' finish with his feet after good work from Wissa and Will Osula in what was Newcastle's best attack of the first half hour.
Barnes broke clear to latch on to Osula's pass but found Leno in his way for a second time, with the German saving with his head. From the resulting corner, which was played short, Murphy crossed for Burn, and the ex-Fulham defender turned home what looked to be the opener - but it was ruled out, with Wissa penalised for a foul on Timothy Castagne at the far post.
Before the break, Horníček made another good stop to deny Josh King, who met Alex Iwobi's cross with venom on the volley from the edge of the box before glancing wide in stoppage time.
Just after restart, Lewis Hall's whipped cross narrowly evaded both Osula and Wissa. The latter then teed up Barnes, whose first-time shot was just off-target as the Magpies rallied.
A low drive from Sander Berge, dragged just beyond Horníček's right-hand post, was a reminder of the threat posed by the hosts, who had won two of their last three games. But Newcastle, perhaps fuelled by what they may have felt was an injustice after Burn's strike was chalked off, kept coming forward and a counter-attack - led by Barnes, Murphy and Osula - resulted in Wissa finding the sidenetting after being forced slightly wide.
Joe Willock and Lewis Miley were introduced by Jaissle just after the hour as United looked to gain the upper hand, and Dedić's cross from the right got away from the stretching Osula before the fit-again Matias Fernandez-Pardo was sent on in place of Sean Steur.
Almost immediately, Berge tumbled under Burn's challenge in the box, but referee Rob Jones was unmoved. The game was there to be won; Leno spilled a Willock shot with 12 minutes left, but recovered just in time to prevent Osula snaffling the rebound.
And in the 80th minute, after picking up a stray Fulham pass in the middle of the park, Murphy advanced and cut in off the right flank before planting a superb shot past Leno's flailing right arm and into the far corner.
But it was a lead they couldn't hold on to, as Cottagers replacement Muniz bundled home a leveller after something of a scramble on the edge of the six-yard box.
It was a bitter blow for the Magpies, who could still have taken the points after that setback. Hall took aim with his unfavoured right foot with two minutes left, but Leno made an important intervention for his side to keep it out.
Horníček, too, made his presence felt at the death, saving brilliantly from Ryan Sessegnon's fierce shot in added time as the spoils were shared.
Newcastle United: Lukáš Horníček, Lewis Hall, Nico González (Lewis Miley 61), Yoane Wissa (Joe Willock 61), Will Osula (Anthony Elanga 85), Harvey Barnes, Malick Thiaw (Fabian Schär 85), Sean Steur (Matias Fernandez-Pardo 74), Jacob Murphy, Dan Burn (c), Amar Dedić
Subs not used: Nick Pope, Tino Livramento, Sven Botman, Bazoumana Touré
