Otago closed play at 232-9, with Ian Butler at the crease unbeaten on 53, while No 11 Blair Soper was yet to score.
The visitors are 244 runs behind on the first innings, and face a huge task in the next two days to avoid a substantial defeat.
Milne registered figures of 5-44 from 12 overs, while import Peter Trego picked up 3-52.
Milne's form comes at a good time, with New Zealand due to tour South Africa next month and the ageing Chris Martin appearing to fall out of favour with the national selectors. He hasn't been given a start in tests on the current trip to Sri Lanka.
At the Eden Park Outer Oval in Auckland, Colin Munro stroked his third first-class century to ensure the home team held the smallest of first-innings leads over Wellington by the close of play on day two.
Munro finished the day unbeaten on 122 - his knock included 14 fours and five sixes - as Auckland closed play on 381-5 in response to Wellington's first-up effort of 380.
Craig Cachopa was the less aggressive of the Auckland pair to finish the day; unbeaten on 74 from 145 balls at stumps.
Earlier in the day, Auckland were in a bit of strife at 174-5 as the top order all made starts but failed to kick on past 50. Tim McIntosh, Lou Vincent, Reece Young and Colin de Grandhomme all made scores between 31 and 46.
Wellington will need to take early wickets tomorrow (Wed) to avoid facing a large first-innings deficit.