Only just over 11,000 were in attendance at Liberty Stadium on Sunday as a section of Swans fans boycotted the fifth-round tie in protest against the club's board.
But those present were treated to an entertaining clash that was turned on its head inside 17 second-half minutes as the home side recovered from falling behind to Ollie Watkins' strike in the opening period.
Bees goalkeeper Luke Daniels' unfortunate own goal levelled the scores before James produced a sensational solo run to put Swansea in front.
52' | GOOAAALLL! What a turnaround! @Daniel_James_97 picks up the ball on the edge of the #Swans' box following a Brentford free-kick and makes a fine solo run before firing past Daniels.— Swansea City AFC (@SwansOfficial) February 17, 2019
Match Centre LIVE! https://t.co/AvVpvvl7ej
2 - 1 | #FACup pic.twitter.com/228talhwA6
The visitors subsequently saw Ezri Konsa sent off before Celina added a fine individual third and George Byers completed the scoring in the final minute, ensuring Swansea equalled their 2017-18 run to the last eight of the competition.
It was a rapid counter-attack that provided the breakthrough for Brentford shortly before the half hour as Said Benrahma and Neal Maupay worked the ball to Watkins, who rifled an emphatic finish into the top-left corner.
But two quickfire goals early in the second period turned the match on its head, Swansea enjoying some luck for the equaliser when Celina's free-kick, won by James, struck the base of the left post and then the back of Daniels before creeping in.
There was nothing fortunate about the second, however, as James charged down Maupay's pass on the edge of his own box before racing the full length of the field and calmly slotting beyond Daniels.
Konsa was played into trouble by Yoann Barbet's poor pass and got his marching orders for bringing down James, before Celina weaved his way past Sergi Canos and Julian Jeanvier en route to a virtuoso goal.
Connor Roberts saw a fourth controversially ruled out for offside almost immediately afterwards but it mattered little as James and Celina combined again to set up Byers as the Swans secured their place in Monday's quarter-final draw.