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Mills claimed 5-38 in Canterbury's second innings - to go with his 5-109 from their first dig - as the seamer and left-arm spinner Bruce Martin (5-45) shot out the visitors for 120.
Only Dean Brownlie offered any major resistance for the Cantabs as he struck 39 and no other player managed to score more than 19.
Canterbury's low second-innings score meant Auckland were in need of 219 for victory and openers Lou Vincent and Tim McIntosh went about knocking off the runs in rapid fashion this afternoon as they put on 71 for the first wicket.
Vincent, a former international who played 23 tests and more than 100 one-dayers for New Zealand, slashed 43 from 28 balls to point Auckland in the right direction.
Once Vincent departed, McIntosh and Reece Young picked up the initiative before Young was sent back to the pavilion for 31 after he and McIntosh had added another 54.
Auckland need only 66 runs from the fourth and final day tomorrow with eight wickets in hand with McIntosh (51 not out) and Craig Cachopa (18 not out) in the middle.
Canterbury will need to produce something miraculous with the ball tomorrow if they want to bring themselves back in to the contest.
In Dunedin, Otago import Ryan ten Doeschate smashed his 19th first-class hundred in his 100th match as the Volts piled on 427 in their first innings on day two of their match against Central Districts.
Rain has threatened the game constantly at the University Oval and there was a stoppage today but it didn't disrupt the flow of ten Doeschate who finished with 118 before he was run out by Jeet Raval.
Otago also received decent contributions from Sam Wells (63), Mark Craig (63) and Neil Wagner (68), while spinner Ajaz Patel was the best of the CD bowlers with 4-86.
Central Districts openers Jamie How and Raval then went about racking up a decent first-innings score of their own and the pair put on 107 for the first wicket before Raval departed late in the day for 47.
Carl Cachopa joined How at the crease and the pair closed day two with Central on 115-1 when bad light stopped play with Cachopa on eight and How on 52.
Meanwhile, no play was possible on day two of the clash between Wellington and Northern Districts at Karori Park due to poor weather in the capital.
Wellington will begin day three tomorrow on 32-4 with Grant Elliott and Scott Kuggeleijn at the crease and the pair will have a big job ahead of them to ensure the home side can avoid a hefty defeat.
Northern Districts set themselves up well in the match when they made 403-8 declared on day one yesterday. (mon)