Stanley had failed to score in their four previous games but they looked the most likely to break the deadlock in the opening half.
Mark Ellis cleared Michael Richardson's shot off the line with just eight minutes gone and Crewe keeper Steve Phillips dived to deny Luke Joyce's 30-yard ripper.
Richardson then headed over a Michael Liddle cross before Seamus McCartan fired high from the edge of the box.
Aneke sliced Crewe's only first-half opening wide after turning Tom Aldred on the six-yard box.
After the break, Alex striker Max Clayton fired into the side-netting before a superb last-ditch tackle prevented Ajay Leitch-Smith from pulling the trigger with the goal at his mercy.
Former Stanley favourite Lee Molyneux then replaced Anthony Grant and let rip with a 40-yard effort which whistled over the bar.
Crewe broke the deadlock in the 79th minute when Aneke rose to nod home Matt Tootle's cross at the near post.