Smith hit his eighth century in 13 tests over the last 12 months, and his fourth in five tests as captain, while western Australian Voges delighted the WACA crowd with his first on home soil to take the hosts to 254 for two.
The skipper will resume on 127 not out and Voges unbeaten on 101 in a partnership of 208 that steadied the innings after New Zealand had removed openers Joe Burns (0) and David Warner (24) cheaply.
That drove home the tourist's advantage after Taylor's mighty 374-ball knock, which featured 43 fours, had dragged his side back into the match with a first-innings lead of 65.
Although he fell 10 runs short of the triple century, the 31-year-old's innings was the highest score by any visiting batsman in Australia, bettering the previous record of 287 set by England's Tip Foster in Sydney in 1903.
It was also the third highest innings by any New Zealand batsman in a test match and eclipsed the 253 contributed by Warner to Australia's first innings tally of 559-9 declared, which had looked almost unassailable on day two.
MAJOR BREAKTHROUGH
Just before lunch and after more than nine hours at the crease, Taylor miscued a slog-sweep to substitute fielder Jonny Wells off the bowling of Nathan Lyon to bring an end to New Zealand's innings on 624.
Breaking Australia's opening partnership, which had put together century stands in the first three innings of the series, was an achievement in itself but also removing Warner with just 46 runs on the board was a major breakthrough.
Southee had Burns caught at first slip by Taylor, while Warner, who also scored centuries in both innings in Brisbane, fell to Trent Boult and a sharp catch from Tom Latham at cover.
Smith came out at number three because of the hamstring injury to Usman Khawaja, who will miss Australia's next two tests, and quickly showed his desire to get in on the WACA run-fest, which has seen 1,441 scored on the first four days.
He raced to his half century in 57 balls and, after being dropped by New Zealand wicketkeeper BJ Watling on 96, reached his 12th test century with a four behind point, the 13th of his innings.
Voges, who hit a century on debut against West Indies earlier this year, scored at a more pedestrian pace and reached the milestone with his 15th boundary in the penultimate over of the day.
(Writing by Nick Mulvenney in Sydney; Editing by Julian Linden/John O'Brien)
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