Bath centre Ojomoh, a late call up for his second cap on Friday, got the first try and set up the second for Immanuel Feyi-Waboso as England led 17-3 at halftime on Sunday.
But Argentina, who came back from 21-0 down to beat Scotland 33-24 last week, fought back again to get within a point, only for Ojomoh to send Henry Slade over.
Rodrigo Isgro scored for the Pumas in the 79th minute to set up a grandstand finish, but England managed to mess up a Puma line-out on their own five-metre line and escaped.
It was England's least convincing performance of the series after wins over Australia, Fiji and New Zealand, but still marked a first sweep of four autumn wins since Eddie Jones' first year in charge in 2016.
The game lacked the electric charge of last week's victory over the All Blacks with relentless kicking and few cohesive attacks, but it certainly provided late drama.
"I understand what my role is and I was waiting to take it onto the field today. I gave a lot of energy in the week, and I'm glad I got to bring it to the field," said man of the match Ojomoh.
England drew first blood with a George Ford drop goal and soon moved 10-0 ahead. Argentina failed to deal with a high Ben Spencer kick and Ojomoh grabbed the loose ball and gleefully galloped over.
Ojomoh, son of former England flanker Steve, then turned provider with a perfect crossfield kick into the arms of Feyi-Waboso.
England were denied a third try when Luke Cowan-Dickie was ruled to have lost possession as he reached for the line with the last play of the first half. The Pumas' only first-half points came from a long Tomas Albornoz penalty.
But poor England defence opened the way for Justo Piccardo to score Argentina's first try four minutes into the second half, the centre finishing off a move he started with a darting break.
Another Albornoz penalty cut the gap to four points after 50 minutes and, with England firing blanks, Steve Borthwick sent on his four-man 'bomb squad' of forwards.
But Argentina continued to dominate and closed to within a point with a Santi Carreras penalty.
England finally built some momentum, entering the Argentina 22 for the first time in the half in the 63rd minute and claimed a superb third try when Ojomoh somehow got a pass away off the floor for his previously anonymous midfield partner Henry Slade to score.
Joaquin Oviedo had a try ruled out by the TMO with two minutes to go, but the Pumas' spirit was undented as Isgro cut through in the 79th minute, leaving just enough time for the restart.
Argentina kept probing and forced a lineout on the England five-metre line, only for the hosts to disrupt it, gather the loose ball and hoof it off the pitch to a roar of relief.
