The world number 11 hit three centuries against the four-time champion including a run of 129 in frame six.
After not potting a ball for more than 40 minutes, Higgins regrouped to snatch the final frame of the session after Allen, 31, missed a difficult green.
The two remaining sessions take place on Saturday morning and evening.
Allen hit breaks of 71 and 65 in the opening frame before a couple of superb pots enabled him to post a run of 100 in the second.
Higgins, 41, fought back to level at 2-2 helped by a break of 70 in frame four but was then shut out entirely over the next three frames as Allen produced the best form of this year's tournament so far with runs of 102, 129 and 82.
Allen looked set to take a 6-2 lead as he moved 30 ahead in the eighth but lost position and Higgins showed batting spirit to snatch the frame helped by a break of 54.
Higgins is likely to be relieved to be only two adrift heading into Saturday's morning session but Allen should be full of confidence after producing probably his best ever Crucible Theatre session.
Northern Ireland's Mark Allen produced a superb first session to take a 5-3 lead over John Higgins in their second-round match at the World Championship.