The triple blow left the hosts - who finished bottom in last season's Plunket Shield - heading into day three trailing by 83 runs with only seven wickets in hand.
Earlier, Redmond (133) stroked his 11th first class century as he and Rutherford (91) shared in a partnership of 186 for the first wicket to set Otago up for a significant lead.
But the Otago middle order failed to capitalise on the platform - with No 10 Ian Butler's 34 the next highest score - as Tim Johnston celebrated his 22nd birthday with figures of 5-102, his best bowling in his fifth first class match.
Elsewhere, Mathew Sinclair began the domestic season doing exactly what he has done throughout his 17-year career - scoring centuries.
The 36-year-old notched his 35th first class hundred as he led Central Districts to a commanding total on the first day of their encounter with Wellington in Napier.
After openers Jeet Raval and Jamie How were both dismissed for 11, Sinclair came together with Carl Cachopa to put on 224 for the third wicket. Cachopa eventually fell for 105 - his fourth ton in the last four first class matches in which he has batted - but Sinclair remained unbeaten on 135 at stumps.
He and Kieran Noema-Barnett (21no) will resume tomorrow as last season's runners-up attempt to build a big total and get off to the perfect start in this year's competition.
Last year's champions Northern Districts and Auckland are not in action during the first round of matches.