Cucurella was sent off by referee Peter Bankes after the full-time whistle, meaning he will miss the Blues' trip to Everton next Sunday.
He will be available to face Shamrock Rovers in the Europa Conference League on Thursday, though.
Cucurella can then return to league action against Fulham on Boxing Day.
Prior to stoppage time, Cucurella had not even been cautioned by Bankes before a manic three minutes ensued for the defender.
After being booked for a foul on Fabio Carvalho, he was shown a second yellow at full-time.
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Cucurella's second booking came for a shove on Bees forward Kevin Shade as he received his first Premier League red card.
The PGMOL said in a statement: "The referee issued a second yellow card to Cucurella for adopting an aggressive attitude. Schade was also booked for the same offence."
Chelsea manager Enzo Maresca commented on the sending-off post-match and revealed what officials told him.
"I didn't see the second yellow card," Maresca admitted.
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"The referee said the second yellow card was for bad attitude. There are things we can do better."
The red card followed Cucurella's first goal of the season to open the scoring in the closing stages of the first half.
That strike was also the 26-year-old's first Premier League goal for Chelsea in his 60th league appearance.
Meanwhile, it was his first in the competition since scoring for Brighton against Manchester United in May 2022.
Maresca was impressed with his left-back's goal and performance, saying: "It's all about the position. You have to be in the right position and if you are you'll have a chance.
"We worked with Marc in this game to arrive and be in the right position. We are very happy.
"It's not something normal because he's not a big full-back. We worked in the week for this kind of solution and we've been lucky."
Nicolas Jackson was the scorer of the second for Chelsea before Bryan Mbuemo pulled a goal back for Brentford.
The win sees the Blues close the gap to Premier League leaders Liverpool to just two points.
However, the Reds have a game in hand following last weekend's postponement of the Merseyside derby.