Coombs and co also face former Heineken Cup winners Bath and French Top 14 team Bordeaux-Begles in Pool 2, with the English giants providing their second round opponents at The Rec on Saturday, October 19.
And while Coombs admits travelling to the West Country will be a particularly tough prospect, he says his side will set off in high spirits if they can complete an Italian double over the coming weekends.
"Things have been positive so far early on in the season so the boys are excited and confident going into the Amlin," said Coombs.
"We've got Mogliano up first at home, which is a nice fixture to start with. It's definitely a chance to build some momentum. We played them twice last season and had two convincing wins against them. I'm sure they would have improved in a season but it's a nice fixture to have first up.
"We've got to play well to beat them but hopefully we can be going down to Bath with a win under our belts and confident to take them on.
"Bath are a big European side and it will be tough to beat them on the road but we've got to play them at home as well. And first and foremost, we've got to win our home games."
Winning at Rodney Parade is something the Dragons have made a habit of this term, with a hard-fought victory over last season's RaboDirect PRO12 table toppers Ulster backed up with an equally impressive success against the Scarlets.
A third straight home win against Zebre would stand them in good stead ahead of the visit of Mogliano but Coombs admits the Dragons have to produce similar results on their travels, starting with that cross-border trip to Bath.
"Rodney Parade isn't an easy place to come. The fans make it a difficult time for the opposition," added Coombs.
"The boys enjoy playing at home and we do play well there, but it's about taking that one step further now and learning to play on the road.
"We played Bath away last season in the LV= Cup. It was a great atmosphere, with a packed out stadium, and I think it was on the same day as an England international so it just shows how loyal their fans are. There's no better place to play than down in Bath.
"The Amlin is great. It's a different spin than the Rabo and it's nice to play in a knockout competition where anything can happen. We need to play well in the opening two games, carry that form on, and then hopefully we can get out of the pool."