The Pumas have now lost seven of their last eight games and their 2016 return is four wins and nine defeats. The result of that is that they have slipped out of the top eight rankings - crucial to seeding for next May's World Cup draw.
England had started confidently and led 3-0 but things took a drastic downward turn when Daly was red-carded for recklessly tackling Senatore in the air, flipping the number eight on to his head.
It was England's first red card since Lewis Moody was sent off against Samoa for fighting 11 years ago and immediately triggered a change in approach with any thoughts of a free-running try-fest safely parked.
Argentina winger Juan Pablo Estelles was lucky to escape any punishment soon afterwards when he similarly brought down an airborne May - the key difference being that May landed more safely.
England kept pressing though and a great hit by Farrell dislodged the ball from Juan Martin Hernandez but with the line beckoning, Matias Orlando batted down the final pass and French referee Pascal Gauzere awarded a penalty try and sent the winger to the sin bin.
There had not been a scrum for the first 35 minutes but then there was virtually non-stop scrummaging five metres from the hosts' tryline.
England defended desperately but illegally and eventually had prop Dan Cole sin- binned to drop them to 13.
Argentina refused to take the guaranteed three points and their faith in their favourite weapon paid off when they finally shoved England back and sent Facundo Isa over to make it 16-7 at the break.
Within a minute of the restart it was 16-14 after a brilliant try by Santiago Cordero as the Pumas backs cashed in on their brief two-man advantage with a scintillating 75-metre attack and started dreaming of a second win at Twickenham 10 years after the first.
England needed to take the space out of the game and did so superbly with driving mauls and pick and go tactics that earned another six points from Farrell to edge them eight points clear.
With that cushion England did cut loose and great work by George Ford and Jonathan Joseph sent May over in the left corner to settle the match.
The talking points kept coming, however, as Argentine replacement Enrique Pieretto was red carded for stamping on the head of prop Joe Marler, who was sin-binned for holding in the same exchange.