But Rutherford remained firm and enjoyed a decent partnership with Neil Broom, advancing the Otago cause past 150. Broom soon fell but Rutherford brought up his seventh first class century before becoming Martin's third wicket and departing for a run-a-ball 108.
Just when Auckland seemed set to make further inroads, Bradley Rodden (34no) and Sam Wells (62no) notched a century stand to see Otago through to stumps without further loss.
Elsewhere, a record partnership between Bharat Popli (82) and Mitchell Santner (118) left Northern Districts in control of their encounter against leaders Canterbury.
The pair resumed where they left off on day one, eclipsing the record seventh wicket stand for the province held by Daryl Mitchell and Tim Southee. Once Popli was removed, ending the pair's resistance at 174, ND were eventually dismissed for 482.
But the conditions may conspire to deny victory, with Canterbury's innings being called off on 16-0 and further rain forecast for the Gisborne region.
The day's other game saw the batting paradise that is McLean Park continue its about-turn, with Central Districts bundled out for 161 in reply to Wellington's 172. The visitors then progressed their score to 143-3 at stumps, leaving them with a lead of 154 at the halfway mark.