With his side leading 1-0, the French full-back was sent off after receiving a second yellow card for a needless shove in first-half injury-time.
It was no surprise, then, when Fulham equalised via teen sensation Ryan Sessegnon in the 49th minute.
However, battling Barnsley still pushed hard for the win and seemed to have done enough to earn a point, at least.
Cruelly, though, Fulham captain Kevin McDonald bundled in a scrappy goal in injury-time and - as the hosts battled to rescue - Sessegnon brutally hit them on the break with his second in the 95th minute.
The Championship strugglers now have just one win in 15 games.
There was a minute’s silence beforehand for Patrick Cryne, Barnsley’s revered former owner who is widely regarded as the ‘saviour of the club’ having rescued them in 2003.
A lifelong fan, he died last weekend at the age of 66 after battling cancer for over a year.
In tribute to their former benefactor, fans poignantly bathed Oakwell in light by turning on their mobile phone lights in the 66th minte, singing ‘He’s one of our own.’
However, down to 10 men, they could not muster the win they so desperately craved to honour him.
It was a shame as Barnsley had mainly controlled the first half, going ahead when Liam Lindsay scrambled in a corner on the half-hour - the centre-half’s first goal of the season - creating a couple of other chances and keeping the visitors largely quiet.
However, all that hard work was undone with Cavare’s implosion.
Barnsley’s Stevie Mallan had drawn a foul from Stefan Johansen in the centre circle but, when the Fulham midfielder said a few words to his rival, Cavare marched over and shoved him in the back.
Johansen, ironically, ended up falling into Mallan, clashing heads and ending up in a heap on the ground.
Referee Andy Davies - who had earlier booked Cavare for a questionable foul on Sessegnon as the teenager surged towards the box - had no other option but to brandish another yellow and dismiss the defender.
Fulham quickly made the most of the extra man at the start of the second period, full-back Ryan Fredericks weaving his way into the penalty box at pace before driving the ball goalwards.
Although his effort struck an upright, Sessegnon was on hand to fired in the rebound in the 49th minute.
The 17-year-old left-back is reportedly attracting interest from the likes of Manchester United, Real Madrid and Paris St Germain and it was easy to see why.
This was his sixth league goal since Christmas Day and - when pushed further upfield in the second period, Fulham giving a debut to on-loan Southampton left-back Matt Targett - was a regular menace for the stretched Tykes defence.
Barnsley also handed a debut to their recent signing, full-back Daniel Pinillos, from Spanish club Cordoba, coming on at half-time as unlucky striker Tom Bradshaw was sacrificed following Cavare’s red card.
In fairness, despite the numerical disadvantage, they still competed hard.
Ryan Hedges went close with a driving run through midfield, forcing a save from Marcus Bettinelli, while Kieffer Moore also found space on the break only to slip at the vital moment.
That said, Paul Heckingbottom’s men also needed captain Adam Davies to produce a fine save from Aboubakar - and then came that late double-blow.
Heckingbottom said: “They say you get what you deserve - we didn’t deserve to win.
“Lots of things I was really pleased with and that’s what we’re going to focus on. I’m really pleased with lots of individuals and the team as a group. There’s always things we can be better at.
“Sending offs happen - you still have to see the game out. We’ll focus on what went right, and what individuals did right.
“We know how good Fulham are, everyone is finding that out now in this league.
“Everyone has to take credit for the first half. But you’re never going to go into a game-plan with 10 men. When you’ve got 10 it’s tough.
“There’s reasons why they scored. Forget that third goal - that was us trying to get an equaliser in the dying seconds.
“Before that I couldn’t see them scoring. If it was it was going to be something from range. It was just unfortunate we couldn’t get the win.”