Higgins only established a meaningful advantage late in the evening, capitalising on a series of missed chances from Murphy to inch ahead.
Frames shared early
The opening exchanges of the session were evenly split, both players taking a frame apiece with breaks of 60 and 55. Neither player was able to impose control, with scoring opportunities quickly turning into tactical exchanges.
The third frame proved pivotal. Murphy led with 44 but missed a black off its spot, allowing Higgins to recover to 34 before play turned cagey. Murphy briefly regained control after a snooker on the yellow, but could not finish the job and Higgins eventually engineered a steal after a decisive snooker of his own, moving ahead 10-9.
Murphy responded immediately with an 82 to level before the interval.
Century landmark and continued parity
After the restart, Higgins struck first again through a break of 63, only for Murphy to be gifted an equaliser, a loose red left in the pocket. The Englishman converted with a superb 105, his 100th Crucible century, to move 11-11, the ninth time the match had been level.
For long stretches, there was little to separate the pair. Breaks were exchanged, chances were shared, but neither player was able to build sustained pressure.
Higgins surges ahead
The decisive moment came late in the session, as Higgins produced a crucial 70 to edge in front again, then followed it with a composed 101, his first century of the match.
Murphy continued to scrap but could not close the gap as Higgins closed out the session with a two-frame cushion.
The semi-final now moves into its final session on Sunday afternoon, with Higgins within touching distance of another Crucible final.