Murphy opened the scoring after 11 minutes when he drove at the home defence and slotted the ball beyond Cammy Bell.
Gary Harkins levelled on the half-hour, but within five minutes Murphy was racing on to Higdon’s excellent through-ball before rounding Bell and rolling the ball into the net.
Fir Park boss Stuart McCall handed left-back Zaine Francis-Angol his first start for the club and the 19-year-old wasted little time in demonstrating his attacking threat.
In the opening minute he charged at the Kilmarnock defence before delivering a dangerous cross which James Fowler was forced to clear behind for a corner.
Francis-Angol was involved at the other end after five minutes when he produced a vital challenge on Rory Boulding, just as the former Livingston man prepared to unleash a right-footed strike.
It was the visitors who continued to look the more likely to break the deadlock and after nine minutes Michael Higdon fired a volley from the edge of the box just over with Bell looking beaten.
Motherwell were rewarded for their bright start two minutes later when Murphy collected a loose pass before driving forward before coolly firing the ball beyond Bell.
Higdon passed up a good opportunity to double the visitors’ advantage when he dragged an effort wide of the target following some hesitancy in the Kilmarnock defence and he then flicked on Francis-Angol’s cross to Chris Humphrey who could not keep his close-range effort on target.
The first booking of the match came when Higdon recklessly fouled Harkins and the booking score was quickly levelled when Liam Kelly lunged in on Keith Lasley.
Boulding sent an effort just wide of Darren Randolph’s right-hand post as the home side tried to find their stride and they found themselves level on 29 minutes.
Harkins collected the ball 30 yards from goal and embarked on a mazy run which ended with a fantastic finish past the helpless Randolph.
The home support found their voices following the equaliser but they soon found themselves behind for the second time in the match.
Higdon threaded a fantastic through-ball which Murphy raced on to, driving forward before rounding Bell and slotting into the empty net.
McCall was dealt a blow ahead of Thursday’s Europa League clash with Levante when Simon Ramsden hobbled off five minutes before the break to be replaced by Adam Cummins.
Nicky Law then should have done better when he rose unchallenged to meet Tom Hateley’s cross on the stroke of half-time but failed to trouble Bell.
After a slow opening 10 minutes to the second half Kilmarnock boss Kenny Shiels opted to make a double substitution, new signing Borja Perez and Danny Racchi replacing Matthew Kennedy and Lee Johnson.
It was the away side who continued to look the more likely though and Murphy missed a chance to complete his hat-trick when he blazed wide of the target from a tight angle a minute later.
Henrik Ojamaa was next to go close for the visitors when he combined with Murphy before curing an effort inches past the post from the edge of the area.
A wonderful piece of goalkeeping from Randolph prevented Kilmarnock levelling for the second time just after the hour. Racchi found himself in space before unleashing a goal-bound effort that the keeper somehow parried before gathering at the second attempt.
Higdon could have put the game out of reach when he met Murphy’s cross on the volley with seven minutes remaining, but could not direct his strike on target.
Shaun Hutchinson received a booking for a cynical challenge on Harkins as the Kilmarnock midfielder tried to orchestrate a breakaway.
To the frustration of the majority of the Rugby Park crowd it was Motherwell who continued to be the more threatening of the two sides and only an excellent stop from Bell prevented Fraser Kerr stretching their advantage to two.