Reid, who was on target in Saturday's 4-3 loss at Leeds, slotted in Josh Onomah's centre to double the hosts' lead in the 32nd minute at Craven Cottage
Aboubakar Kamara had put the Premier League side on course for a relatively comfortable evening against lower-league opposition by turning home Anthony Knockaert's ninth-minute pass.
Scott Parker's side will make the short journey to the Brentford Community Stadium in the last 16, a repeat of last month's Sky Bet Championship play-off final.
Wednesday, the last club from outside the top flight to lift the League Cup following victory over Manchester United at Wembley in 1991, produced a spirited display in west London but never seriously threatened to spring a surprise result.
Fulham went into the game on the back of successive Premier League defeats following promotion.
Manager Parker made seven changes for the tie, retaining Denis Odoi, Kamara and former Owls loanees Michael Hector and Onomah from the weekend defeat at Elland Road and allowing star striker Aleksandar Mitrovic a night off.
While the home side's main priority is remaining in the top flight, Wednesday face an uphill battle to stay in the second tier.
The Owls, who began the new campaign on minus 12 points as punishment for breaching financial rules, made a single alteration to the second-string team which started at Rochdale in round two, handing a senior debut to boyhood fan Liam Waldock in midfield.
Monk's men had not conceded in their opening four matches but that impressive run of clean-sheets was brought to an abrupt end.
Knockaert broke clear of the visitors' backline before calmly teeing up Kamara for a simple close-range finish past the stranded Joe Wildsmith.
Wednesday's inexperienced line-up enjoyed the greater share of first-half possession and could easily have levelled.
Reach stung the palms of Rodak from distance with 22 minutes played, while the unmarked Dele-Bashiru later wasted a golden chance by sidefooting straight at the Fulham goalkeeper after good work from Odubajo.
Owls keeper Wildsmith conceded five times on this ground in a 5-3 defeat just two months ago and he was picking the ball out of his net again 13 minutes shy of the break.
Minutes after repelling a powerful drive from Antonee Robinson, the former England Under-20 stopper was unable to prevent Onomah's fizzed delivery from the left being diverted in at the back post by Reid.
Wednesday wing-back Odubajo tested Rodak either side of half-time but the away team faded in the second period and spent most of it attempting to keep the scoreline respectable.
Reid was denied his second of the evening by the legs of Wildsmith after breaking clear.
Knockaert, a player who has caused Wednesday plenty of problems in the past, then skewed wide when well placed, before clipping the top of the crossbar with a speculative chip from distance.
Parker used the final 12 minutes to hand a senior debut to Fabio Carvalho.
The 18-year-old winger, on in place of Knockaert, came close to a dream maiden outing but his lobbed effort landed narrowly wide as Fulham comfortably secured a fourth-round spot for the first time in six seasons.