Former Toon winger Dan Gosling has helped Bournemouth top the league, although he was injured and didn't make the teamj last night.
Midfielder Murphy slammed an unstoppable shot into the top corner and Neil Redfearn's side secured a morale-boosting first win in eight league matches - but only after Bournemouth substitute Yann Kermorgant had missed a late penalty.
Bournemouth, who were twice denied by the woodwork, remain top of the table but slipped to their first league defeat on the road since September 30 and lost for only the second time in 17 Championship matches.
Kermorgant fired his 87th-minute penalty against the crossbar and although Marc Pugh's 81st-minute shot hit a post, Leeds held on for their first win since November 29.
Patrick Bamford and Lee Tomlin scored the goals as Middlesbrough rose to second in the table with a 2-1 win over Cardiff.
After a goalless first half, Chelsea loanee Bamford opened the scoring in the 63rd minute to leave the home fans delighted.
It appeared they would go home happy when Tomlin struck 11 minutes from time, his effort appearing to put the result beyond doubt.
But there looked like a twist in the tale was on the horizon when Kenwyne Jones struck in the 86th minute to set up a tense finale.
It mattered little, however, as Aitor Karanka's men held on the claim a valuable three points.
The Teesside club subsequently rose in the table and now sit just a point behind leaders Bournemouth, albeit with the rest of the chasing pack holding a game in hand.