Craney beat Lambs keeper Scott Bevan from the edge of the area with a low drive after Tamworth had failed to clear their lines in the 56th minute.
The goal proved to be the winner and Stanley were good value for the victory as they dominated the majority of the game against a Tamworth side who have struggled to reproduce their impressive FA Cup form in the league and are languishing in the bottom two.
It took Accrington just three minutes to do what Championship side Stoke City failed to do in 90 minutes last Saturday.
Lambs full-back Michael Touhy made a costly error and Gary Roberts picked out Paul Mullin with a superb cross. Mullin's header beat Bevan and bounced in off the post.
Bevan was by far the more busy of the two keepers and he produced two good saves to deny Craney and was out quickly to save at the feet of Mullin.
Tamworth were given a lifeline in the 39th minute when Michael Welch was penalised after grappling with Jake Edwards inside the area.
Edwards, who was possibly playing his last game for Tamworth after spending two months on loan from Exeter City, stepped up to score his seventh goal in 13 games.
However, Accrington were still the better side and they could have added to Craney's second-half strike but for another superb save from Bevan who managed to tip Mullin's strike on to the post.