Everton secured a 1-0 victory at St James' Park on Monday, with Barry scoring the decisive goal in the 49th minute to give David Moyes' side another important result against the Magpies.
Tarkowski was particularly pleased with the team's first-half performance, believing Everton could have established a more comfortable advantage before the break.
"I think it was probably our best performance of the season, I thought we were terrific in the first half and could have scored two or three," the defender told Sky Sports.
Barry provided the finishing touch after the interval, continuing a productive spell in front of goal. Tarkowski believes the striker's contribution is especially important given the limited attacking options available to Moyes.
"Thierno gets the goal again for us. There's a lot on his shoulders, he's our only recognised striker so hopefully he can keep putting the ball in the net like he has done," he said.
"He's harsh on himself but his numbers are through the roof in terms of running."
The clean sheet was another source of satisfaction for Tarkowski, who stressed that Everton must maintain their defensive standards throughout a long campaign.
"A clean sheet is massive for us. It's a long old season. We take it game by game but tonight feels good and we can look forward to Friday [against Coventry]."
Barry was equally pleased to contribute, explaining that the work he put in during the summer is beginning to produce the results he wanted.
"I just working a lot during the summer and now we can see the result. The season is long but I'm happy to help the team with my goals," the forward said.
The Frenchman also spoke about his celebrations, explaining that he is determined to enjoy his football more after a difficult previous season.
"Like I say, I want to enjoy more than last season and I enjoyed my celebration so I am happy."
Barry also acknowledged the physical challenge posed by Newcastle United's defenders, singling out Dan Burn and Fabian Schär as two of the toughest opponents he has faced this season.
"I think Dan Burn and Fabian Schar were the most difficult defenders I have faced this season because they are really strong but for the striker you just need to score and I won my duels today so I am happy."
The goal was another encouraging moment for Barry, who has had to shoulder significant responsibility as Everton's recognised centre-forward. His willingness to work for the team, combined with an improving goalscoring return, is giving Moyes further encouragement.
For Tarkowski, the performance offered evidence that Everton are making progress, with the defender particularly pleased by their attacking threat and ability to protect their advantage.
The victory also completed a second win over Newcastle in five days, giving Everton momentum as they prepare for Friday's meeting with Coventry.