Playing in their first knockout match on the world stage since the agonising extra-time 2018 semi-final loss to Croatia, Gareth Southgate's men showed their quality to progress in Qatar.
Kane became England's highest ever scorer at major tournaments after Henderson opened the scoring, with Saka superbly wrapping up a comfortable 3-0 win against Senegal in Sunday's last-16 contest at Al Bayt Stadium.
The reward is a return to Al Khor for a crunch clash against France next Saturday as last year's European Championship runners-up take on the World Cup holders.
Southgate's side will need to take their performance up a level given some of the problems Africa Cup of Nations winners Senegal were able to cause in a first half that ended with a quickfire England double.
Henderson was the unlikely man to open the scoring when coolly directing in the cutback from Jude Bellingham, who played a key role as England also scored with the final kick of the first half.
The 19-year-old led a brilliant breakaway that ended with Phil Foden playing on to 2018 Golden Boot winner Kane, who slammed home his first goal in Qatar to become the side's eighth different goalscorer.
That strike also left the prolific Tottenham striker on 52 England goals, one behind the all-time record set by Wayne Rooney.
Saka then joined Marcus Rashford - the man he replaced in the starting line-up on Sunday - on three goals at this World Cup and ended the contest when he delicately clipped the ball over Edouard Mendy early in the second half.
England were in cruise control from that point and Southgate rang the changes with progress assured to consecutive World Cup quarter-finals - something the national team last managed in 2002 and 2006.
Southgate had earlier lost the services of Raheem Sterling, who was absent from the last-16 clash due to a family matter on Sunday, while Senegal head coach Aliou Cisse overcame illness to take his place on the touchline.
Senegal's superb support backed their team with relentless, rhythmic drumming in Al Khor, where England began with patience and composure.
Mendy flapped at a Bellingham cross before Kane fizzed over another ball from the left that Saka could not reach, with John Stones bundling off target.
Senegal settled and were proving an increasing threat, going agonisingly close in the 22nd minute.
Harry Maguire's pass out was picked off by Krepin Diatta, who sent over a cross from the right that Boulaye Dia met with a volley that Stones took the sting out of before Ismaila Sarr lashed over the loose ball.
Handball appeals against Stones followed but a VAR check came to nothing.
England continued sloppily and were soon breathing another sigh of relief. Saka gave the ball away in a poor area and Sarr smartly slipped in Dia to get away a fizzing shot that Jordan Pickford stopped with a strong arm.
Senegal were looking confident and most likely to score, only for England to step up a gear and land the first blow.
Quick play down the left ended with Kane putting through Bellingham, with the teenager cutting back for Henderson to turn a low left-footed strike past Mendy in the 38th minute.
Cisse told Senegal's players to focus and keep their cool as England pushed for a second before the break.
Saka centred for Kane but the skipper lifted over and Declan Rice saw a cross cut out, before England added another goal in stoppage time.
Bellingham collected the ball, burst forward and held off a challenger before playing left to Foden, whose first-time pass took a slight touch but still found Kane.
The skipper continued forward to rifle past Mendy and send the travelling England supporters wild.
England returned from the break with a spring in their step. Saka struck wide and a Kane pot shot was unconvincingly saved by Mendy, who would be beaten again in the 57th minute.
Luke Shaw won the ball and England quickly worked down the left, with Foden riding a challenge and crossing through Kalidou Koulibaly's legs to Saka, who casually clipped over Mendy from 10 yards out.
Rashford and Jack Grealish replaced Saka and Foden as England flexed their strength in depth, with Mason Mount and Eric Dier joining them after Pape Matar Sarr tried to catch out Pickford from a free-kick.
Koulibaly was booked for a foul on Kane, who could not quite stretch enough to reach a Rashford cross after shaking off that knock.
Henderson made way for Kalvin Phillips and Rashford fired into the side-netting in stoppage time, with 'Sweet Caroline' echoing around the ground at full-time.