Rosenborg, managed by Englishman Robin Shroot, upset the odds on Wednesday when they knocked out Atletico Madrid in a dramatic penalty shootout.
They showed they could cause problems and Arsenal vowed not to be complacent as a result, despite going into their second match of the week heavy favourites.
Shroot was keen for his side to express themselves with attacking football, but also make it hard work for Arsenal to break them down.
But the Gunners quickly imposed their authority and Rosenborg were not allowed to settle, rarely getting out of their own half until after the break when they found space.
Summer signing Caldentey was once again busy and entertaining, Maanum a threat and Cooney-Cross dominated the midfield.
It was still at the back of everyone’s minds though, that Arsenal were knocked out at this stage last season by Paris FC, and there were nervy moments late on as Rosenborg threw players forward, desperately trying to create an opening on the break.
But Arsenal looked more composed and sharp this time, albeit against weaker opposition.
Jonas Eidevall’s side know they need to step up another level this season to compete with Women’s Super League (WSL) title rivals Chelsea and Manchester City, and are building promising foundations.
The Gunners will avoid the seeded clubs in Monday's second-round qualifying draw, and have shown plenty to suggest they can cope with their next opponent in a tie over two legs.