Ford converted and then added a penalty, the first five-eighth finishing with a match haul of 12 points.
But Argentina wing Rodrigo Isgro scored a last-minute try, converted by Santiago Carreras, with time still left for a restart only for a botched lineout in the fourth minute of stoppage time, as the Pumas pressed for a match-winning try, allowing England to close out the game.
Defeat ended Argentina’s bid for an unbeaten tour after wins over Wales and Scotland.
England made their first entry into the Argentina 22 count as Ford landed a ninth-minute dropped goal.
And barely 60 seconds later, Ojomoh, playing just his second Test following a debut away to the United States in July, ran in untouched for a try after Argentina failed to deal with a high kick from halfback Ben Spencer.
Ford converted and England led 10-0.
But the Pumas, who had already beaten the British and Irish Lions, New Zealand and Australia this year, then laid siege to England’s line before Carreras, initially on as a head injury replacement for Albornoz, saw a 37m penalty hit the post.
Ojomoh’s superb cross-kick then landed straight into the arms of the unmarked Feyi-Waboso, who sprinted in unopposed for a 25th-minute try, with Ford’s conversion making it 17-0.
The Pumas eventually cut the deficit five minutes before halftime when Albornoz landed a long-range penalty.
Ford then missed a penalty but there was still time in the half for England to win another and this they kicked for touch to set up an attacking line-out.
From the ensuing catch and drive, Luke Cowan-Dickie rumbled over only for the score to be disallowed after a television review found the hooker had knocked on while grounding the ball.
Argentina hit back when Piccardo powered through some questionable tackling for a 46th-minute try, converted by Albornoz.
With England’s lead now reduced to seven points, Argentina coach Felipe Contepomi sent on a raft of replacements.
Argentina then won a penalty in front of England’s posts and the game was back in the balance, with the hosts now just four points ahead and nearly half an hour remaining.
England boss Borthwick responded by freshening up his pack.
Ford missed a 63rd-minute drop goal but an Argentina ruck infringement saw him kick for an attacking lineout won by replacement back-row Henry Pollock.
Quick ball from the set piece was followed by a superb back-of-the-hand pass from Ojomoh that sent in Slade for a try between the posts.
Ford converted and England, who won twice on tour in Argentina in July, were now two scores ahead at 24-16.
Argentina replacement Joaquin Oviedo thought he’d scored a 79th-minute try, only for his effort to be disallowed, just as Cowan-Dickie’s effort had been, for a knock on, before England clung on in a frantic finish.