Four minutes after the interval Jambos midfielder Ian Black was shown a straight red card by referee Stevie O’Reilly for a challenge on Keith Lasley.
The Steelmen capitalised on their advantage on the hour mark when Murphy converted a pass from Nicky Law.
Motherwell should have added to their lead but they ended the game in a nervy state after Lasley, booked in the first half, was sent off in the 86th minute after picking up his second yellow card for a foul on Hearts substitute Arvydas Novikovas, who had replaced Jamie Hamill.
But in the end it was a deserved win for Stuart McCall’s side who leapfrogged Celtic, 1-0 winners at Aberdeen earlier in the day, by one point.
Hearts will lay most of the blame at the feet of the short-fused Black, who left his team-mates a man short for most of the second half and the turf was scattered with prostrate maroon-shirted players when the final whistle sounded.
It was the visitors, however, with Ryan Stevenson supported by David Templeton in attack, who threatened first.
There were only four minutes played when Jambos’ wide-man Andrew Driver robbed the careless Steven Saunders 25 yards out and sent in a drive which Motherwell keeper Darren Randolph did well to push behind for a corner.
The home side defended the set-piece easily enough but it was an early warning.
In the 20th minute Tynecastle keeper Marian Kello pulled off a good save from Motherwell midfielder Steve Jennings, who had picked up a Chris Humphrey cross which had made its way to far side of the box.
Just on the half hour, as Motherwell struggled to clear their lines following yet another inswinging Danny Grainger corner from the right, Jambos captain Marius Zaliukas picked up a loose ball but from 30 yards he drove wide of the target.
Murphy should have put the Fir Park side ahead in the 33rd minute when he met a Humphrey cross at the far post.
However, the highly-rated Well striker appeared to mis-time his jump and headed over the bar from six yards.
As the game grew increasingly tetchy, Lasley was deservedly booked for a foul on Adrian Mrowiec before Randolph had to rush from his goal to deny Driver after a sliced clearance from Saunders put his side in trouble.
Three minutes from the break the home keeper was forced into another save, this time from Hamill’s long-range effort which threatened to sneak in at his left post.
A game that had threatened to boil over eventually did in the 49th minute when Hearts’ volatile midfielder Black was shown a straight red card by referee O’Reilly for a reckless challenge on Lasley in the middle of the park.
The Hearts midfielder took his time leaving the pitch, apparently questioning the validity of the decision but his dismissal left his side playing most of the second half with 10 men and the home fans urging their side to take advantage.
Former Motherwell striker John Sutton replaced Driver in the 57th minute as the visitors reshuffled.
As the rain continued to cascade down, Motherwell upped their efforts in their bid for the opener which came on the hour.
Saunders released Law down the right and when the Well midfielder’s pass found Murphy in the box he took his time but eventually slid the ball past Kello after shrugging off the attentions of Jambos’ defender Ryan McGowan.
The Fir Park side were clearly buoyed by the opener and with some gusto went in search of number two.
Kello was kept busy as the Fir Park side swarmed around his goal and in the 77th minute Hamill was booked for tripping Humphrey as he tried to break in to the Hearts box following a swift counter.
From the free-kick, Tom Hateley’s shot was partially blocked and there was a shout for a penalty when Murphy went to ground following a tackle by Eggert Jonsson, but O’Reilly pointed for a goal kick.
Four minutes from normal time, Lasley was sent packing after picking up his second yellow card for a foul on Hearts substitute Arvydas Novikovas, who had replaced Hamill, but the home side held on.