The East Region Junior side held out for 28 minutes before Murray's close-range finish broke the deadlock.
Paul McManus began a second-half goal deluge when he netted from six yards for the League Two leaders.
Murray grabbed his second after being teed up by Kevin Nicoll and added his third in the 75th minute before McManus found the bottom corner a minute later.
The part-timers had a chance to give the majority of the 1,769 crowd something to cheer late on only for Stuart Hunter to head over.
Match ends, Bo'ness United 0, Arbroath 5.
Second Half ends, Bo'ness United 0, Arbroath 5.
Foul by Kevin Buchan (Arbroath).
Ross Campbell (Bo'ness United) wins a free kick in the attacking half.
Craig Johnstone (Arbroath) wins a free kick on the right wing.
Foul by Nicky Walker (Bo'ness United).
Attempt missed. Darren Gribben (Bo'ness United) right footed shot from outside the box is close, but misses to the right.
Foul by Ali El-Zubaidi (Arbroath).
Mark Cowan (Bo'ness United) wins a free kick in the defensive half.
Scott McBride (Arbroath) wins a free kick in the defensive half.
Foul by Ian Nimmo (Bo'ness United).
Foul by Mark Whatley (Arbroath).
Andy Scott (Bo'ness United) wins a free kick in the defensive half.
Attempt missed. Stuart Hunter (Bo'ness United) header from the centre of the box is too high.
Substitution, Arbroath. Kevin Buchan replaces Paul McManus.
Corner,  Bo'ness United. Conceded by Kevin Nicoll.
Foul by Adam Hunter (Arbroath).
Nicky Walker (Bo'ness United) wins a free kick in the defensive half.
Substitution, Arbroath. Jack Smith replaces Simon Murray.
Goal!  Bo'ness United 0, Arbroath 5. Paul McManus (Arbroath) right footed shot from the left side of the box to the bottom right corner. Assisted by Bobby Linn.
Goal!  Bo'ness United 0, Arbroath 4. Simon Murray (Arbroath) right footed shot from the left side of the six yard box to the centre of the goal. Assisted by Craig Johnstone.
Attempt missed. Paul McManus (Arbroath) right footed shot from the left side of the box misses to the right.
Foul by Simon Murray (Arbroath).
Stuart Hunter (Bo'ness United) wins a free kick in the attacking half.
Foul by Bobby Linn (Arbroath).
Mark Cowan (Bo'ness United) wins a free kick in the defensive half.
Goal!  Bo'ness United 0, Arbroath 3. Simon Murray (Arbroath) right footed shot from the centre of the box to the bottom right corner. Assisted by Kevin Nicoll.
Goal!  Bo'ness United 0, Arbroath 2. Paul McManus (Arbroath) right footed shot from very close range to the bottom right corner. Assisted by Simon Murray.
Attempt missed. Scott McBride (Arbroath) header from the centre of the box is just a bit too high.
Darren Gribben (Bo'ness United) is shown the yellow card.
Ricky Little (Arbroath) wins a free kick in the defensive half.
Foul by Darren Gribben (Bo'ness United).
Substitution, Bo'ness United. Mark Cowan replaces Kieran Anderson.
Attempt missed. Craig Johnstone (Arbroath) right footed shot from outside the box misses to the left.
Corner,  Arbroath. Conceded by Stuart Hunter.
Substitution, Bo'ness United. Andy Scott replaces Chris Donnelly.
Corner,  Arbroath. Conceded by Mark Peat.
Attempt saved. Bobby Linn (Arbroath) left footed shot from outside the box is saved in the bottom right corner.
Corner,  Arbroath. Conceded by Ross Philp.
Attempt missed. Kieran Anderson (Bo'ness United) right footed shot from the right side of the box is just a bit too high.
Arbroath's Simon Murray grabbed a hat-trick to end Bo'ness United's cup dream in an emphatic win at Newtown Park.