Walcott struck five minutes after the restart, keeping his composure in a congested penalty box to power a shot into the top corner for his first goal in nine matches.
Newcastle struggled to create clear-cut equalising chances although they could have had an 87th-minute penalty when the ball arguably struck Jagielka on the arm.
Relive how the action unfolded on Merseyside with Sports Mole's minute-by-minute updates below.
Subs: Robles, Martina, Funes-Mori, Baningime, Davies, Calvert-Lewin, Niasse
Subs: Darlow, Clark, Manquillo, Hayden, Merino, Murphy, Gayle
Jordan Pickford is also in a battle with Jack Butland to establish himself as England's new number one.