Murray and Wilfried Zaha were the star men at Selhurst Park, the striker bagging two goals and one assist as the winger impressed with two assists.
Championship top goalscorer Murray made his mark after nine minutes, tapping home after good work from the excellent Zaha as Palace acquitted themselves well against the team just one place below them.
The former Brighton forward opened the scoring early on, but missed two more chances to complete a first-half hat-trick as Palace struggled to make their dominance count.
Though Murray scored a brace, Zaha endeavoured to steal the show as England manager Roy Hodgson watched on, keeping an eye on the Manchester United-bound starlet.
Ian Holloway's men continued to dominate the game with defender Peter Ramage scoring a second, while Murray smashed home his sublime second and Palace's third after 57 minutes.
Middlesbrough striker Faris Haroun combined well with Ishmael Miller to break through the Palace backline and slot home after 79 minutes, but the goal was only a consolation for a poor Boro side.
Boro were restricted to long-range efforts as Julian Speroni saved shots from Andrew Halliday and Mustapha Carayol.
New signing Stephen Dobbie failed to connect with Murray's cross before the former Swansea midfielder let Boro off the hook by opting to shoot rather than pass in a promising position as Palace laboured towards victory.
But Kevin Phillips starred as a substitute for promotion-chasers Palace, latching on to Zaha's through-ball and finishing well on 84 minutes to secure all three points for the Londoners.
Palace were rarely troubled by Tony Mowbray's men, who have now lost six of their last seven games in the Championship to put them in danger of dropping out of the coveted play-off positions.
Palace, meanwhile, sit 12 points behind leaders Cardiff, but nine clear of seventh placed Brighton.