The hosts riding on Vaughany's majestic effort posted 319 all out in their first innings in reply to Kiwis first innings total of 277.Vettori, who had reduced England to 208 for six with three quick wickets, finished with five for 69, including the wicket of Vaughan for 106.
It was his first five-wicket haul against England, his 14th overall.
At the close of the first full day's play in the match, New Zealand had reduced the deficit to two when they reached 40/0 in their second innings.
Jamie How and Aaron Redmond were at the crease with 26 and 14 respectively.
Andrew Strauss (63) and Alastair Cook (61), resuming at 89 for no wicket after 153.2 overs had been lost to rain and bad light on the first three days, recorded their first century opening partnership for England.
Their departures were the signal for left-arm spinner Vettori to make his mark on the match, bowling for the first time after having two stitches to a split on his bowling finger removed before the match.
Vettori accounted for the dangerous Kevin Pietersen, lbw for three, then removed Paul Collingwood (6) and Ambrose (0) from consecutive balls in his 10th over.
Broad narrowly avoided edging the hat-trick ball and survived some anxious moments to add 61 runs for the seventh wicket with Vaughan before he was bowled off his pads by Jacob Oram for 25.
Vaughan, who averaged only 22.53 against the Kiwis compared to a career figure of 42.47 before the match, drove sweetly through the on-side and greeted Vettori's return after tea with consecutive fours through mid-off and over mid-on.
He cut Vettori to the third-man boundary for four and late-cut another three off the spinner who up to that time had been treated with the utmost respect by the England batsmen.
Vaughan reached his century by flicking Vettori through mid-wicket for his 11th four to equal former England captain Graham Gooch's record for Test centuries at Lord's.
He was finally out with last man James Anderson at the crease, caught on the mid-wicket boundary by James Marshall trying to hit another boundary.
Vaughan gave only one chance on 64 when Redmond at second slip was unable to hold on to a difficult chance low to his right hand off paceman Tim Southee.