Having failed to win in their last four qualifiers, the Republic required three points against Moldova in order to set up a decisive clash with Wales on Monday, and the Boys in Green duly obliged with a dominant performance in the capital.
The hosts settled any early nerves inside two minutes, as Murphy reacted quickest inside the area in order to turn a finish beyond Ilie Cebanu, securing only his second international goal for the Republic.
The opener seemed to dispel any home nerves and Ireland soon set about their attacking work, this time as Callum O'Dowda broke away on the counter to put Shane Long through on goal, but the striker dragged his effort wide of the post.
Securing a safety net was Ireland's next challenge and they put themselves into a comfortable position in the 19th minute, Murphy grabbing his second of the night with an outstanding header to cap off a memorable move between Wes Hoolahan and Stephen Ward.
Moldova offered very little in the opening 45 minutes as they sat deep inside their own half, although Sergiu Platica almost produced a moment of magic with a dipping volley that saw Darren Randolph rising to tip the ball over his crossbar.
Seeking to add a third before the interval, the Republic nearly had centre-back Ciaran Clark to thank, but the defender somehow sent his header wide of the post with just Cebanu to beat.
The ROI arguably came out in the second period lacking purpose and they allowed the visitors back into the game for a short period, although Eugeniu Cociuc could not muster the required quality despite getting the better of Ward.
After missing a standout chance in the first half, Long had the opportunity to redeem himself and put the game firmly to bed after the hour mark, yet he was again wasteful after side-footing wide when O'Dowda's initial shot was blocked.
The miss failed to make much difference to proceedings, though, as Moldova were reduced to 10 men late on when Alexandru Gatcan saw red, leaving the Republic to stroll home and establish a mouthwatering clash with Wales next week, the winner of that contest claiming a definite finish in the top two of Group D.