Tommy Seymour threw an unnecessary pass and Murray pounced to score.
Scotland lost Stuart Hogg to injury, and then went further behind after a brilliant set backline play.
A planned move unleashed Jacob Stockdale and he raced away from 45 metres out to touch down.
Ireland took a commanding 12-3 lead in the 17th minute.
But Scotland, after Joey Carberry replaced Sexton, hit back through Russell. The fly-half intercepted Carberry’s pass and offloaded to Sam Johnson for a quality try.
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Greig Laidlaw converted to make it 12-10.
Ireland withstood huge pressure right before half-time to keep their two-point buffer at the break.
In the second half the visitors found a second gear to close out the match.
It was the sub Carberry who burst through the middle and set up Robbie Earls for the try in the corner.
In the 62nd minute Laidlaw cut it back to 19-13 with a penalty goal.
But with 12 minutes left Carberry nailed a penalty of his own to take it to 22-13.
Scotland pushed for a response but their handling let them down. The hosts made 14 errors to just seven for Ireland.
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The victory moves Ireland up to fourth spot on the Six Nations table with four points.