Tottenham showed there is life after Harry Kane with a 2-0 win over Manchester United on Saturday as Manchester City beat potential Premier League title challengers Newcastle 1-0.
Spurs were playing for the first time at home since record goalscorer Kane departed for Bayern Munich.
"Second half, certainly in spells, it kind of showed the team we want to be," said Postecoglou.
"I keep saying we're still a long way to go but I really liked the belief and resilience the guys have and the bravery they have to continue playing that way."
Erik ten Hag's men escaped with a 1-0 victory at home to Wolves on Monday despite being outplayed.
This time United were punished as they faded after a bright start.
Bruno Fernandes should have headed the visitors in front midway through the first half.
And Postecoglou's men got their reward when Sarr smashed home his first goal for the club on 49 minutes.
A comedy of errors summed up United's day for the second goal as Ben Davies failed to connect with Ivan Perisic's cross and Martinez could only turn it beyond the flat-footed Andre Onana.
United's decision not to rival Bayern with a bid for Kane will now face more scrutiny as they looked toothless without new striker Rasmus Hojlund due to injury.
"I am not concerned with our attacking play but I am by our scoring," said Ten Hag. "We know this, that is why we've signed a striker."
- City show their class -
Many believed Newcastle would never get a better chance for their first league win away to City since 2000.
Pep Guardiola was unhappy the champions had been forced to play just three days after winning the UEFA Super Cup over Sevilla in baking conditions in Athens.
City were also without Kevin De Bruyne, Bernardo Silva and John Stones, but were still far too good for the Magpies.
It took one moment of magic for Guardiola's men to take all three points, though, when Julian Alvarez fired into the top corner on 31 minutes.
Erling Haaland missed a hat-trick of chances to add to the lead, but City were rarely troubled defensively and made it two wins from two Premier League games.
"It is the second game of the season, but it is proof of the reason why we won a lot," said Guardiola.
"Today the team showed to me and the Premier League we are ready to try to win it again."
Liverpool recovered from a terrible start and the controversial loss of Alexis Mac Allister to a red card early in the second half to beat Bournemouth 3-1.
Antoine Semenyo drilled the visitors in front after just three minutes on the ground where they lost 9-0 last season.
Luis Diaz's inventive flicked and overhead kick finish brought Liverpool level before Mohamed Salah made it 2-1 on the rebound after his penalty was saved.
Momentum could have swung back Bournemouth's way when Mac Allister was harshly shown a straight red card for leading with his studs on Ryan Christie, despite minimal contact with the Scotland international.
But the 10 men added to their lead within four minutes when Diogo Jota pounced after Neto could only palm Dominik Szoboszlai's effort into his path.
- Seagulls soar -
Brighton banked a British record transfer fee that could rise to £115 million ($146 million) for Moises Caicedo on Monday.
But the Seagulls continue to soar under Roberto De Zerbi despite seeing their best players picked off by the Premier League's elite.
Kaoru Mitoma may be the next big-money departure and the Japanese international opened the scoring in stunning style at Molineux as he slalomed past four Wolves defenders before producing a classy finish.
Mitoma then teed up Pervis Estupinan to smash home for 2-0 before Solly March scored twice from Julio Enciso assists.
Brentford are also making light of the absence of the suspended Ivan Toney as they won 3-0 at Fulham.
Yoane Wissa opened the scoring before Bryan Mbeumo scored twice after Fulham captain Tim Ream was sent off for a second yellow card.