Catalysis of the reaction: a beta-D-Gal-(1->4)-beta-D-Glc-(1<->1)-Cer(d18:1(4E)) + UDP-N-acetyl-alpha-D-glucosamine = a beta-D-GlcNAc-(1->3)-beta-D-Gal-(1->4)-beta-D-Glc-(1<->1)-Cer(d18:1(4E)) + H+ + UDP.
UDP-N-acetyl-D-glucosamine:D-galactosyl-1,4-beta-D-glucosylceramide beta-1,3-acetylglucosaminyltransferase activity