The Royals had bagged seven in the thrashing of Bolton Wanderers last week but despite that and a suspect defence on their own soil, Mick McCarthy's side recorded a narrow victory.
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Stephen Hunt had the two clearest chances of the first half but was twice denied his opening Town goal by Reading defender Kasper Gorkss, who blocked a tame volley following a cross from Luke Chambers.
He then cleared an effort from the Republic of Ireland international off the line, with Hunt guiding a Murphy knockdown goalwards, but his compatriot was more fortunate in front of goal after the break.
A thumping drive from Murphy went in off the underside off the crossbar on 62 minutes to put Ipswich on their way to a victory which takes them to within two points of Reading in the last playoff place.
Hunt was involved heavily until his late withdrawal and was perhaps lucky to escape without a red card when he was cautioned for sliding in late on Royals keeper Alex McCarthy.
Substitute Anderson put the result beyond doubt with a fine individual effort in stoppage time, ghosting past his marker before powerfully firing across McCarthy and into the side-netting.