The 26-year-old scored his side's opening goal against Cyprus on Monday night to extend the Dragons' unbeaten run to three games in qualification for Euro 2016.
Having regained confidence in recent times, Wales are now in a strong position to reach a major finals, and Cotterill believes that a lot of the positive team spirit comes down to former boss Speed.
"He started all this off, the togetherness," the Birmingham City winger told BBC Sport. "Every time we've come away the last few camps everyone seems together and it definitely changed when Gary Speed took over.
"Chris Coleman's taken that on board as well and things are improving. Everyone here is working extremely hard for each other and we're all together."
Wales return to action next month with a trip to face Group B favourites Belgium.