Steve Borthwick's side outscored Los Pumas three tries to two, with Max Ojomoh, Immanuel Feyi-Waboso and Henry Slade crossing for the hosts. George Ford added 12 points from the tee, including a drop goal.
Argentina stayed in the fight through tries from Justo Piccardo and Rodrigo Isgro, while Tomás Albornoz and Santiago Carreras kept the scoreboard moving with penalties and conversions.
England led 17-3 at half-time after Ojomoh intercepted for the opening try and then set up Feyi-Waboso with a cross-kick.
But Piccardo's try early in the second half sparked an Argentine comeback, and Carreras' penalty narrowed the gap to 17-16 heading into the final quarter.
England struck again when Ojomoh broke the line and offloaded for Slade to score, giving the hosts the cushion they needed.
Ford's late penalty pushed the lead to 27-16 before Argentina grabbed a consolation try in the dying moments.
A late defensive stand sealed the result, ensuring England's winning streak stayed intact.