The visitors led at half-time at Tottenham Hotspur Stadium via a scrambled Neal Maupay finish.
Ivan Toney pulled one back following a dreadful Udogie backpass, but Spurs held on to win and move back into the top four for the first time in 2024, after Ange Postecoglou's side dropped from the Champions League places following the 4-2 defeat by Brighton on 28 December.
It was a frenetic game with 28 shots across both teams in which Spurs were drawn into squabbles with Brentford during the first half before pulling away in the second with an early blitz.
"It's a young, maturing team," Postecoglou told BBC Sport. "Friday [the FA Cup defeat by Manchester City] showed we have a long way to go and tonight was another test. Arguing with the opponents isn't going to help us."
Spurs started the game on top, but were caught out with their high defensive line as Toney was played through. His shot was saved by Gigi Vicario, but Maupay pounced on the rebound.
Although the Frenchman kicked the ball against his standing foot, it deflected into the corner to give Maupay his fourth goal in consecutive matches.
But Spurs came out with a bang after half-time, scoring three times in eight minutes. First, Udogie raced forward from left-back and exchanged passes with Timo Werner, before finishing at the second time of asking after his first shot was blocked.
Seventy-three seconds later Werner was the provider again with a low cross from the left finished from close range by Johnson with just his second touch after coming on as a half-time sub - before Richarlison completed the goal rush with the follow-up after Maddison's shot was deflected.
Richarlison has now scored seven goals in his past seven league appearances, one more than he managed in his previous 47 top-flight outings.
Spurs allowed Brentford back in the game in farcical circumstances with Udogie playing a backpass straight to Toney, who calmly slipped the ball past Vicario for his second goal in as many matches since returning from his eight-month ban for betting offences.
But Brentford could not find an equaliser and remain 15th, just four points above the relegation zone.
Maddison 'sharpens his tools' on Spurs return
Much of the focus before the game was on James Maddison, who made his first start since 6 November following his return from injury off the bench against Manchester City.
Maddison was crucial in Spurs' electric early-season form, and his last start before injury came against Chelsea at home - their first league defeat of the season.
Spurs fans were hoping the England midfielder could inspire them back into the top four, and a jinking run through the defence after 10 minutes - before being crowded out in the area - gave hope he could make the difference in attack.
But Spurs' comeback after the break came when Maddison was moved into a deeper role, with Johnson introduced at half-time in place of Oliver Skipp as part of a double substitution.
While Maddison did find himself in the area to inadvertently assist Spurs' third, the change of shape allowed Werner and Johnson to flourish out wide.
"He's been out for a while - today he showed why we missed him so much," Johnson told Match of the Day. "He has the X-factor. Today was him sharpening his tools again."
Their fast start to the second half extends Spurs' club-record run of now scoring in 34 successive league matches.
Although they still have clear flaws - particularly in defence, as showed in second-half stoppage time when they allowed Shandon Baptiste space in the box for a volley which Vicario tipped over acrobatically - Spurs are certainly stronger with Maddison in the side, whatever his role.
Brentford rue '10-minute blackout'
Brentford played Tottenham perfectly in the first half, both in exploiting their high line - Mads Roerslev finished a slick move minutes before Maupay's opener, only for Mathias Jensen to be caught offside in the build-up - and in getting inside their heads.
Maupay was central in needling the Tottenham players, encapsulated by him replicating Maddison's darts-throwing celebration following the opening goal.
His celebration aggravated some of the home supporters and Maddison himself, who shared an unfriendly exchange of words and head pats with Maupay afterwards.
It clearly riled Maddison's team-mates too, with both Johnson and Richarlison pulling out the darts celebrations after their goals as Brentford capitulated in a 10-minute spell which manager Thomas Frank rued.
"We frustrated them with the high press, we were good with the low block, on the break and transition we were very dangerous," Frank told Match of the Day.
"Then we had a blackout, so it is about being smarter. The second goal is the most disappointing - don't concede from the kick-off."
The Bees have now dropped a league-high 23 points from winning positions this campaign, and have lost five league games this season after scoring the opening goal, more than any other side in the division - and once more than Tottenham.