Lancaster earned all three points against Macclesfield thanks to first-half goals from Sam and Charlie Bailey.
Sam Bailey gave the Dollies the lead from the penalty spot after he was fouled in the area.
Then Charlie Bailey's sweetly-struck half-volley on 42 minutes put City two goals to the good at half-time.
Macclesfield had several chances before they did pull a goal back from Rooney's spot-kick.
The home side then struck the post and Dollies 'keeper Andre Mendes pulled off a late save to ensure City took a vital win.