The former test opener crunched five sixes and nine fours in a destructive knock of 110 from 107 deliveries to help Otago post 213 for seven declared.
Set 400 for victory, Central made a terrible start with Neil Wagner cleaning out Ben Smith with a yorker which clattered into middle stump.
The visiting side will resume this morning on seven for one with Otago in the ascendancy but all three results still possible.
Rutherford batted aggressively throughout his stay but was particularly brutal on left-arm spinner Ajaz Patel, swinging consecutive sixes on to the embankment at midwicket. Another was sent sailing towards the sightscreen and pinged from the bat as he closed in on his second hundred of the season and 10th of his career.
``I wanted to score reasonably quickly to try and put us in a position where we could dictate,'' Rutherford said.
With 37 overs lost on day two, there was a pressing need to get on with it and Rutherford's innings has helped buy back some time.
``Initially, we were hoping for a lead of 300 but to add another 100 is pleasing. And 400 on this wicket will be quite tough.
``There is more than 100 overs [today] so there is enough of a carrot for them to try and chase it down. But also for us there is enough assistance out there to take nine wickets.''
Central Districts claimed the honours in the morning session, adding 112 runs for the loss of two wickets.
The visiting side resumed on 80 for four in reply to Otago's first-innings total of 402.
Dane Cleaver swung away some big shots but, having reached 50, tried to whip Christi Viljoen into the legside.
Neil Broom thrust up an arm and snatched the ball from the sky at a short mid-on position. It was a brilliant piece of fielding from the veteran and broke a 112-run stand between Cleaver and Tom Bruce.
Bruce had proved adept at hitting the ball into gaps and milked the third man boundary when it was vacant.
But with lunch in sight he hovered back into the crease and was trapped in front by Finnie.
His departure for 83 exposed the lower order and Otago ran through the rest of the line-up.
Volts keeper Derek de Boorder pounced with a classy stumping and a fine catch up at the wicket.
Doug Bracewell made a useful 42 not out but ran out of support as the visiting side collapsed to be all out for 216.
Boasting a generous first-innings lead of 186, Rutherford and Brad Wilson batted with freedom.
The team 100 came up off 113 balls and was the pair's third 100-run stand this season.
But in the search for quick runs, Otago lost some cheap wickets at the end and it was Patel who benefited.
Rutherford danced down the wicket looking to dispatch him for another six, got a wide delivery and was stumped.
Finnie holed out at long on in Patel's next over, and Anaru Kitchen cut the next ball straight to Blair Tickner. The hat trick ball passed harmlessly by.
The late wickets were a bit of a sideshow, though, with Otago's lead already quite large.
Meanwhile, Canterbury needs 78 runs to beat Northern Districts but is running out of wickets. It is 130 for seven in what shapes as a close finish at Seddon Park.
The match at Eden Park Outer Oval is seemingly heading for a draw. Wellington is 99 for two in its second innings and leads by 200 runs with a day to play.
