Newcastle's victory ended a run of three consecutive defeats which had seen Howe under the spotlight.
Howe said he "didn't see" the incident which prompted referee Craig Pawson to award Newcastle a penalty, but Brentford manager Thomas Frank strongly disputed it.
"We have a keeper clearly pulling out and a player cleverly letting his leg hang there," Frank told BBC Sport.
"I'm convinced that VAR would not have overturned the decision, so I think that's a problem. But because it's not clear and obvious, it's too tight.
"Now we will have a situation where Howard Webb will come out and apologise about the situation and say 'we're sorry'. Going forward these should not be penalties."
Howe heads to San Siro in better spirits
The chewing gum might have been gnawed more intently, but otherwise there was little in the sanguine Howe's touchline demeanour in the technical area to indicate he was feeling the pressure.
Victory over a Brentford team unbeaten in seven league games before this encounter helped lift the Magpies to 11th place as they notched their second win of the season.
In keeping with his persona, there was no outpouring of emotion from Howe at the final whistle, his mind doubtless already on Tuesday's Champions League opener at San Siro against AC Milan - who were thrashed 5-1 in Saturday's Milan derby.
In attempting to ignite their season Howe made five changes to his starting line-up, but Brentford's enterprise was the more polished in the early stages.
Hickey went close while Yoane Wissa was unable to apply the decisive touch to Mathias Jensen's fizzing delivery shortly after. What the Bees lacked was a cutting edge in attack as the ongoing absence of Ivan Toney - suspended until January for betting offences - was acutely felt again.
Newcastle finished the first half the stronger and Bruno Guimaraes saw his diving header at the back post saved by the feet of Flekken.
Predatory Wilson underlines his qualities
The home side rediscovered some of the intensity which had helped propel them into the top four last season in the second half - and Wilson was again the difference.
The 31-year-old signed a new contract this week which will potentially extend his stay on Tyneside until the summer of 2025.
His predatory instincts in front of goal were once again underlined as he enabled Howe's side to win a match slim on clear-cut chances.
Alexander Isak had been given the task of spearheading Newcastle's attack in their opening four matches, but after a poor run of results Howe turned to experienced goalscorer Wilson.
The England striker was unfortunate to have been deemed to have fouled Flekken when he turned the ball home from close range shortly after the break.
However, when Flekken brought down Gordon the Newcastle number nine stepped up and emphatically buried the spot-kick.
"He is a leader in the group, a vocal presence, a positive guy and a goalscorer," Howe said.
"Put them together and he's very important, so I'm very happy he's staying with us."
Wilson has scored his past 11 penalties, a run stretching back to May 2021, and now has 14 goals in his past 17 appearances having only started seven of them.
He should be a shoo-in to be in the starting line-up at San Siro next week.
Howe added: "It was a bitty game for us. We had to get set-pieces right, crosses right and defensively I was very pleased."