Wu was not put off his stride by a brief power outage at the Crucible that held up the match after the 22-year-old had taken a 2-0 lead.
Allen hit back to level at 2-2, but Wu was on course for a maximum break in the fifth frame until missing his attempt on the eighth black to restore his lead.
Further half-century breaks of 78 and 64 opened up a 5-2 lead before he held off an Allen fightback in the final frame of the evening.
Earlier, Higgins hit back from a 3-1 mid-session interval deficit to leave a repeat of the 2009 final finely poised.
The 50-year-old Scot, playing in his 100th match at the Crucible, started slowly after a late night in beating Neil Robertson in the quarter-finals.
But Murphy could not drive home his early advantage with two further sessions to come on Friday.
The first player to 17 frames will progress to the final, which begins on Sunday.
