Academy chief Murphy, 63, watched from his usual seat in the directors' box as on-loan Barnsley frontman Hayes headed in Andy Taylor's third minute cross.
Bradford defender Richard Edghill should have cleared Taylor's centre but he failed and Hayes pounced to score his first goal for Town on his third appearance.
Hayes then sprung the City offside trap with a through ball for Danny Schofield to score on 75 minutes and ensure relegation-threatened Bradford's bad run stretches to only one win in 14 games.
Hayes could even have had a second when Bantams caretaker boss David Wetherall miscued and allowed Hayes to bare down on goal only for keeper Donovan Ricketts to save with his legs.
Murphy, in temporary charge following Peter Jackson's sacking on Tuesday, could be set for a longer stint at the helm than first thought as the Terriers have been denied permission to talk to their top target.
Gillingham chairman Paul Scally is believed to have told Gills boss and Huddersfield promotion hero Ronnie Jepson that he cannot discuss the vacant Town job with opposite number Ken Davy.
Hayes' early strike should have set up an exciting derby contest but it was a scrappy affair with few chances at either end for the 14,772 lunchtime crowd to savour.
The closest Bradford came to a goal was on 39 minutes when Moses Ashikodi, on loan from Watford, beat the Town offside trap and raced towards goal.
The 19-year-old Nigeria-born England starlet seemed certain to score but was stopped by a brilliant recovery tackle by Andy Holdsworth.
Four players from each side were booked, but apart from a melee involving several more late on, it was not a dirty game.