A much-changed Celtic side had the better of the SPL clash but struggled to get past Darren Randolph before the influential Shaun Maloney won a penalty from Steven Saunders.
Saunders was sent off and Motherwell failed to recover from the blow as Murphy stroked home the 73rd-minute spot-kick.
Neil Lennon made five changes to the team that were dumped 4-0 in Holland on Thursday as he handed debuts to Fraser Forster and Emilion Izaguirre.
On-loan Newcastle goalkeeper Forster had very little to do while Honduras left-back Izaguirre looked agile in defence and attack.
Glenn Loovens, Maloney and Murphy also came in and Efrain Juarez, Jos Hooiveld and Beram Kayal dropped to the bench, with Marc Antoine-Fortune having left for West Brom since Celtic threw away their two-goal lead to exit before the group stages.
That defeat was the third cup clash Celtic have lost since Lennon took over but he continued his perfect SPL record with an 11th consecutive victory against a Motherwell side who looked jaded following their European exit against Odense, when they did everything but score in a 1-0 loss.
Both sides had great chances to open the scoring inside three minutes. John Sutton met Tom Hateley's inswinging free-kick but headed just wide and Randolph pulled off a brilliant one-handed save to deny Maloney from close range following James Forrest's cross.
Maloney's free-kick then just cleared the bar after the winger had incurred the wrath of Stephen Craigan for going down on the edge of the box.
Randolph made another impressive save to tip over Murphy's deflected effort and the former Charlton keeper then held Forrest's 20-yard drive.
Maloney and Forrest's advanced wide positions gave Lennon's formation the look of a 4-2-4 and the visitors were pushing Motherwell back, with the wingers combining for Maloney to head wide of the near post.
The home side stemmed the tide for the second quarter and began to enjoy some good spells of possession without ever getting in behind the Celtic defence.
Motherwell left-back Stevie Hammell went off with a hamstring injury in the 35th minute with youngster Jonathan Page coming on.
They continued to look comfortable but the only time they threatened was when Loovens sliced a clearance over the head of Forster but beyond the far post.
Maloney almost made the breakthrough in the final seconds after beating Hateley Saunders inside the box but he shot into the side netting.
Motherwell were forced into a half-time change when Steve Jennings, who had been struggling with illness was replaced by Nick Blackman and Well moved to a five-man defence.
Celtic soon created two decent opportunities but Daniel Majstorovic headed over from Izaguirre's corner and Randolph got down well to hold a measured strike from Georgios Samaras from 20 yards after a forceful run by Scott Brown.
Celtic were again pushing Motherwell back and Paddy McCourt replaced Samaras midway through the half shortly before Forster made his first save, a comfortable one from Blackman's shot.
Maloney continued to be at the heart of Celtic's threat after moving inside. His clever pass set up Murphy, who shot wide from a tight angle and the former Aston Villa player shot inches wide of the far post following McCourt's run.
Maloney then won the penalty as the Motherwell defence were caught square just inside their own half following Randolph's goal kick as the forward burst clear on to Majstorovic's defensive header.
Saunders led the chase but got too close and Maloney went down before Dougie McDonald pointed to the spot and showed the defender a straight red card to add to the yellow he had received in the first half for catching Samaras late.
Murphy sent Randolph the wrong way to find the top corner and Celtic held out comfortably while threatening to add a second.
McCourt came close on several occasions as the Motherwell defence backed off and also set up Murphy, who finished poorly from inside the box.