Ireland, who led Group D early on before three points in their previous four games pushed them back to third place, trail Wales by a point ahead of the decider against their near neighbours in Cardiff on Monday.
Wales' victory in Georgia earlier on Friday meant nothing but three points would do Ireland and Martin O'Neil's side got off to the perfect start when Stephen Ward's long throw caused panic in the box and Murphy opened the scoring after less than two minutes.
If Moldova could have defended better for the first goal, they could do little for the second 17 minutes later when a sweeping Wes Hoolahan pass found Ward whose cross was met by the retreating Murphy, looping a long-range header over Ilie Cebanu.
Shane Long should have twice extended the advantage and at the other end Sergiu Platica almost scored one of the goals of the qualifiers when his left-footed half volley from outside the box was acrobatically tipped over by Darren Randolph.
Ireland, who had only scored twice at home in their previous four qualifying games, were unable to add to the advantage as the Modovans almost made a game of it in the second half.
But they never seriously came close to threatening the hosts' goal before Alexandru Gatcan was sent off in injury time for a head-butt as Irish thoughts turned to Cardiff and the hope of being one of eight best runners-up.
"We knew what we needed to do in this game and happily we did it," Ireland's Callum O'Dowda, who impressed in his first international start, said in a pitchside interview.
Both Ireland and Wales could still technically top the group after leaders Serbia failed to wrap up automatic qualification following a 3-2 loss to Austria. They remain one point clear of Wales ahead of their final game at home to Georgia.
(Editing by Rex Gowar)
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