Arsenal rode their luck on occasion before taking a 42nd-minute lead through Gervinho, only to be pegged back minutes later when Kostas Mitroglou glanced home a header.
The home side needed just 11 minutes of the second half to re-take the lead, though, when Lukas Podolski netted, and a fine late finish from Aaron Ramsey sealed a second successive victory in Champions League Group B.
Bould, who again sat on the bench in place of Arsene Wenger as the Frenchman continued to serve a European ban, said of the victory: "It was tough, they made it tough for us. We started well enough but then looked a little bit nervous.
"Overall it was fantastic result."
With Schalke being held by Montpellier at home this evening Arsenal are now two points clear at the top of the group, although Bould is not getting carried away.
He told Sky Sports 4: "It's early days, there's an awful lot of work still to do."
Captain Thomas Vermaelen echoed those thoughts, admitting his side were not up to speed on occasion in the first half before improving after the break.
The centre-back said "The second half was better, in the first half we were sloppy sometimes with the passing, we lost a lot of balls.
"We stared well but the first half was not great."
Vermaelen did not think Arsenal's 2-1 home defeat to Chelsea at the weekend had any influence on their sluggish first-half showing on Wednesday night.
He said: "We started well, with a high tempo and that was great but then we dropped a little. I don't think it was because of the weekend, sometimes these things can happen.
"We did much better in the second half."
He added of the win: "It gives us more confidence to have two wins from two. The most important thing is to win your home games. We want to keep it going."
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