Bremer converted a Douglas Luiz cross midway through the first half, and Juventus dominated large spells without adding to their lead, missing several clear chances before a late Frosinone rally almost forced a leveller, with Gabriele Calvani's header cannoning back off the crossbar in stoppage time.
Spalletti was satisfied with the identity his side showed despite the missed opportunities.
"Overall I think we played a good match, some players very well, others well, and maybe without foreseeing situations that could have given us further advantages, but these are normal situations. The important thing was to start and show we're a team, and we did that.
"There'll be time to improve. Frosinone played a great match, the game I expected. I thought we might struggle more, but we were good. Credit and respect must go to the opponents too."
Asked whether the display underlined the need to complete the squad before the transfer window closes, Spalletti left the door open.
"Certainly the directors will be active and will assess what happens during the week, but we could also stay as we are. We're a strong team all the same, even stronger if situations like today aren't finished off. The directors will be there from early morning to see what comes up. We're not in a position to say we want this and just do it, you have to find solutions where the possibilities line up with the negotiation."
Pressed on why he didn't substitute Randal Kolo Muani despite a missed chance, Spalletti defended the forward.
"Kolo had a good match, why take him off? He needs to get back into the rhythm of the league. It's been two years since he's played regularly and he needs to find match rhythm. A mistake is understandable, and this missed goal is one he owes us back. Beyond the missed chances, there were three or four situations where we could have shot but didn't manage it."
On Khéphren Thuram's fitness, he added: "Thuram is better, he's started training with us, but his condition is still fragile, even if he feels fine in the small-sided games. He still needs the work."
Frosinone coach Massimiliano Alvini felt his side's response after a sluggish opening spell deserved credit.
"Frosinone started well, then had less effectiveness, we had a section of the first half that wasn't good, we didn't run well. We waited a bit too much, which isn't like us. Juventus deservedly went ahead and could have finished it off, then we closed things down well. In the second half we were effective, we could have equalised and it would have been important for us. Overall Juventus deserved the win. I say well done to the team for the effort and desire shown."
Asked whether the performance meant he needed reinforcements before the window closes, Alvini declined to be drawn on the market.
"I don't want to talk about transfers, it's the first match of the league and it was fantastic. We competed with Juventus right until the last second, even forcing Juventus to adapt to us at times. The transfer market doesn't depend on me, I don't want to talk about it. I'm happy with the performance against a great opponent."