Wells eventually fell for 79 while Rodden was the penultimate man out, having compiled 118 during a six-hour stay at the crease.
The onus was on Auckland in the final session to score quickly if they wanted a positive result, but regular wickets interrupted their charge and left them on 175-5 at stumps.
The hosts' second innings got off to the worst possible start when Anaru Kitchen was trapped in front for a duck by Neil Wagner in the first over, while fellow opener Jeet Raval didn't last for much longer.
Craig Cachopa (70) and first innings centurion Colin Munro (49) enjoyed a handy partnership for the third wicket but both were dismissed by Mark Craig, while Colin de Grandhomme's stay in the middle was short-lived. Gareth Hopkins (10no) and Donovan Grobbelaar (8no) were unbeaten at the close of play and the pair will have to accelerate the scoring tomorrow morning, of which both men are more than capable.
Elsewhere, Northern Districts' worst-against-first clash with Canterbury remains in the balance, despite much of the third day in Gisborne being lost to rain.
That fact owes to a sporting declaration from Peter Fulton (78no) who, together with Tom Latham (56no), added 139 runs without loss before calling a close to the innings. Northern Districts ended the day on 5-0 to hold an overall lead of 348.
Rain also disrupted Central Districts' match against Wellington in Napier, with the visitors able to add only 112 to their overnight score to lead by 266 with six wickets in hand heading into the final day.