Sturridge, who has failed to score in seven Premier League matches this season, has only found joy in front of goal in England's second-tier knockout competition when top-flight typically deploy weaker teams.
Sturridge's double eliminated Tottenham with a 2-1 victory at Anfield, his first Liverpool goals since also netting twice in the League Cup in August.
After firing into the roof of the net from Marko Grujic's cross in the ninth minute, the England striker tucked the ball under Tottenham goalkeeper Michel Vorm in the second half.
It was a smoother night for Arsenal against Reading, despite only beating the second-tier League Championship club 2-0.
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Oxlade-Chamberlain poked the ball through defender Tyler Blackett's legs into the corner of the net to put Arsenal ahead in the 33rd minute. A deflected shot from the England international doubled Arsenal's lead in the 78th.
Premier League side Hull survived a late scare against Bristol City of the League Championship to win 2-1.
Harry Maguire and Michael Dawson scored either side of halftime but City forward Lee Tomlin grabbed a goal in the third of seven minutes of stoppage time.
The fourth round will be completed on Wednesday with three all-Premier League matches, headlined by United hosting City in the season's second Manchester derby clash between Jose Mourinho and Pep Guardiola. There's also a London derby between West Ham and Chelsea, while Southampton takes on Sunderland.