Gianluca Scamacca's first-half opener was followed by the dismissal of Gus Ledes on the stroke of half time, before Jarrod Bowen netted a three-minute brace at the start of the second half to become West Ham's all-time leading scorer in European competition.
Eighteen-year-old Divin Mubama capped off a comfortable night in the capital with his first senior goal in the 65th minute to help the Hammers extend their winning run in the competition to 10 matches.
David Moyes has guided West Ham into back-to-back European quarter-finals for the first time in more than half a century, and the Hammers will have their name in the hat for Friday's last-eight draw.
West Ham entered the contest having to cope without injured striker Michail Antonio, who scored both goals in the 2-0 first-leg win in Cyprus last week, with under-fire Italian Scamacca tasked with leading the line and improving his individual form in front of goal.
The £40m summer signing from Sassuolo, who was starting his first game since January 4, made a bright start to the second leg and had goalkeeper Kenan Piric sprawling early doors with an audacious strike from outside the area on the left flank.
For all their dominance, the Hammers lacked a cutting edge in the final third until Manuel Lanzini played through Scamacca for a somewhat scrappy opener in the 20th minute.
West Ham's No.7 received the ball inside the area and cut inside before poking a right-footed effort through the legs of Piric, who did not cover himself in glory.
Scamacca thought he has scored a second just three minutes later, but the Italian was unable to beat the offside trap before stroking the ball into the bottom corner.
Larnaca then had the ball in the net on the half-hour mark when Nemanja Nikolic fired beyond Alphonse Areola, but the 35-year-old was marginally caught offside.
Pablo Fornals came agonisingly close to doubling West Ham's lead when he rattled the underside of the crossbar with a curling strike in the 36th minute.
Larnaca's hopes of a miraculous comeback were then quashed on the stroke of half time when Ledes' yellow card for a high tackle on Fornals was upgraded to a red following a VAR review.
The Portuguese midfielder was sent for an early bath for the second time in four days after seeing red in Larnaca's 1-0 Cypriot First Division defeat to Apollon on Monday.
West Ham flew out the blocks in the second half with Bowen scoring twice within the first three minute, slotting his first in from a tight angle after Piric parried a strike from Scammaca on the edge of the area.
The Englishman then doubled his tally with a brilliant looping header, directing a whipped cross from the left by Aaron Cresswell over Piric and into the far corner.
Bowen has now surpassed teammate Antonio as well as club icons Johnny Byrne and David Cross to become West Ham's all-time leading scorer in European competition with eight goals.
West Ham's No.20 was unable to complete his hat-trick, though, as he was replaced before the hour mark along with Scamacca and Lucas Paqueta, with Moyes opting to bring on Danny Ings, Maxwel Cornet and academy graduate Mubama, the latter handed his first senior appearance at the London Stadium.
The academy graduate looked sharp as soon as he came on, and the teenager was rewarded with his maiden goal for his boyhood club in the 65th minute, watched on by his family in the stands.
An out-swinging corner from Cresswell was headed into the six-yard box by Tomas Soucek and Mubama was on his toes to flick the ball home to the delight of the London Stadium faithful.
Larnaca understandably offered little going forward after going down to 10 men, and West Ham comfortably saw out the tie to book their place in Friday's quarter-final draw.
West Ham will return to action after the international break when they play host to basement club Southampton in a crucial relegation battle in the Premier League, while Larnaca will continue their push for the Cypriot top-flight title when they face Aris Limassol next Monday.