The home side had reached 270 for seven by stumps on day three and nurses a modest lead of 165.
Wagner sliced through the top order, reducing Wellington to 35 for three.
But James Franklin led the fight-back with a dogged century - his 19th first-class ton - and shared in a 146-run stand with Michael Pollard (82).
Earlier, the Volts resumed on 341 for eight, with Derek de Boorder unbeaten on 116 and Craig Smith on nine.
Smith was able to add only a further five runs to his overnight tally but contributed to a valuable 55-run stand for the ninth wicket which helped Otago add to its first innings lead.
De Boorder was the last man out. Experienced medium-pacer Brent Arnel nicked him out for a career-high 146.
It was de Boorder's third first-class century and would have come as a relief.
He has struggled with the bat this summer and changed his back lift earlier in the season to avoid falling into the lbw trap.
Otago No 11 Jacob Duffy did his best to get de Boorder through to 150.
He saw off 26 deliveries in his nought not out.
Having conceded a first innings deficit of 105, Wellington stumbled early but recovered through Franklin and Pollard.
Duffy secured a double breakthrough, removing Pollard and then bowling Tom Blundell for the third-ball duck.
Franklin lost another valuable aide in Luke Woodcock for 17, and Matt McEwan for five.
Franklin will resume on 108, with Jeetan Patel at the wicket with him on 18.
Wellington has a lot of batting ahead of it today if it is to get to safe ground.
In Rangiora, Auckland strengthened its grip on its match with Canterbury.
The visitors posted 491 in reply to Canterbury's first innings total of 268.
At stumps on day three, Canterbury was struggling at 154 for seven. It trails by 69 runs and defeat appears inevitable.
In New Plymouth, Central Districts held on for a draw against Northern Districts.
Asked to follow on having conceded a first innings deficit of 154 runs, Central had reached 268 for five when both captains agreed to an early end.
Northern's opener, Dean Brownlie, will have found memories of the match.
He became the seventh New Zealand batsman to post a first-class triple century.
