Havertz continued to burnish his reputation with a first-half brace as Leverkusen crushed lowly Werder Bremen 4-1 in the Bundesliga on Monday.
The Germany international – frequently the topic of transfer gossip columns amid links to Liverpool, Manchester United and Arsenal – opened the scoring in the 28th minute and restored Leverkusen's lead five minutes later following Theodor Gebre Selassie's equaliser.
After thrashing relegation-threatened Bremen behind closed doors at Weser-Stadion due to the coronavirus pandemic, Bosz told reporters: "Kai [Havertz] is a special player, but I don't think he made his best game today. He can do much better.
"I've known him for a long time now and I know he can do much better. It has to do with the rhythm. But for every player the captain's armband is something special. And so it will be for Kai."
Havertz, 20, added: "I'm very happy. We didn't make many mistakes and deservedly won.
"Now we want to continue like this on Saturday in Monchengladbach."
The result was Leverkusen's fourth in succession as the Bundesliga returned over the weekend following the COVID-19 outbreak.
Leverkusen are fifth in the table, eight points behind defending champions and leaders Bayern Munich.
"We are naturally satisfied, as you have to be when you win the first game after a two-month break with 4-1 in Bremen," Bosz said. "There were many question marks before. I think we deserved to win, although we didn't always play very well. That has to do with the rhythm.
"We lost easy balls during the game, which we normally don't lose. But like I said, if you win 4-1 here, you have to be satisfied. And I am satisfied."
The Bundesliga returned in empty stadiums on Saturday after a two-month hiatus due to the coronavirus pandemic, with Hertha recording a 3-0 victory at Hoffenheim.
However, while precautions were taken in the games - including mask-wearing substitutes sitting significant distances apart from one another - there was no social distancing being observed when Dedryck Boyata appeared to plant a kiss on the cheek of Marko Grujic after one of the goals.
The German Football League (DFL) has confirmed it will not sanction players for celebrating, though it has offered recommended guidance on the matter, and Bosz wants players to avoid letting emotions get the better of them.
"In the beginning we talked about emotions and they are part of football," he said at his news conference to preview Leverkusen's trip to Werder Bremen on Monday.
"I can imagine that when Hertha scores at Hoffenheim, that they want to celebrate. They must not do it this way. I know that.
"Next time, I don't think they'll do it that way. But goals are emotional moments in which they may have forgotten.
"You're not allowed to do that, but I can imagine that it happens from time to time."
Bruno Labbadia, who was taking charge of his first game at the Hertha helm, had suggested that the frequency of testing meant his players should be allowed to celebrate in such a manner.
"Celebrating goals is part of football. We have been tested so often that I think you can allow it," Labbadia said.
"It would be a shame if you weren't allowed to celebrate any more.
"I hope people out there understand it. It is just a recommendation [from the DFL] to hold back.
"We have tested negative six times, most recently on Friday. Emotions are part of the game, otherwise we don't need to play it."
The 23-year-old has settled down well to life at the German club after missing the start of the season, due to enforced quarantine measures in his home country. Since returning to the line-up, however, Bailey has shown solid form. Not only has he proven to be elusive for defenders but he has also managed to notch four assists. Two coming in the team's last two wins.
With a good portion of the previous season impacted by injury and indifferent form, the player will be happy to be rounding back into the form that saw him labelled as one of Europe’s brightest prospects.
"Well, Leon had it very difficult, last year and at the beginning of this season. But I can see that he gets better every game, physically but also in terms of football,” Bosz said.
"When you see how much he runs, but also how much he sprints and the quality of his football game, that always goes up. Last game in Nice the first goal he set-up for Nadiem Amiri was really great. And he played well today too. He improves every game. "
Havertz scored the winning goal as Leverkusen moved up to third in the Bundesliga by edging Freiburg 1-0 away from home.
The in-demand Germany international, who made it five goals in four games since the coronavirus-enforced break ended this month, hobbled off just past the hour with a knock but Bosz allayed the injury concerns.
"He is fine. I don't know how bad it is, but I think he's gonna be okay," Bosz said after the match.
It was a record-breaking outing for Havertz, who has been linked with the likes of Real Madrid, Liverpool, Bayern Munich and Manchester United.
The 20-year-old midfielder scored his 35th Bundesliga goal, making him the first player under the age of 21 to reach that total.
"I'm very happy for him, of course," Leverkusen team-mate Leon Bailey told bundesliga.com. "He's very, very talented - I think we all know that."
"For him to be able to be doing that gives young players a vision," Bailey added. "Anything is possible once you work hard. I think Kai Havertz is a very hardworking player.
"He's smart, and very intelligent on the pitch. I hope he continues to do this and help the team as much as possible because we need him."
