An England team without 15 British and Irish Lions upheld its record of never losing a series in Argentina since the first tour in 1981.
Saturday's match in San Juan was headed for a 17-17 draw until, with a minute left, Argentina spilled possession on their 40-metre line and England counterattacked.
Replacement back-rower Guy Pepper burst into the open, ran over or drew the last three defenders and gave backup scrumhalf Van Poortvliet a clear run to the tryline.
George Ford couldn't convert from the sideline, missing his first goalkick off the tee in the two Tests, but the tour captain was rapt with his young team's success against a Pumas team which beat the Lions three weeks ago.
"We found a way in the end," Ford said. "We had a lot of opportunities in the 22 but we just couldn't execute. I like our fight, our togetherness, showing how much we care for each other. We should have scored more tries but the attitude of the group is great."
After a ding-dong first half in which the lead swapped four times, Ford's tying penalty was the only score in the second half until the end.
England spent most of the second spell in Argentina's half but the Pumas hustled and scrambled in defence.
Overall, England had 17 visits in Argentina's 22 in the match but scored only three tries. But it was just enough after hammering the Pumas 35-12 in La Plata last weekend.
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