The 18-6 win, their fourth in a row, lifted the Scottish club to fourth and they are now threatening top spot with five wins from eight following a poor start.
It also condemned the Italian outfit, whose only triumph this campaign was against Edinburgh,to their fourth straight defeat, but Lineen insists their self-control was the secret to their success.
"These games we would have lost in the past," he said. "To really nail it just before half-time, getting a really important drop goal to go from 9-6 to 12-6, shows a little bit of maturity and patience.
"These games away in Italy are difficult to play because they are so physical, but we showed a lot of composure in key periods over a number of areas, and I am just really happy for them.
"Our scrummaging went really well, which was a real surprise over here in terms of the strength they have so all credit to the full props, who all did a marvellous job.
"We are working incredibly hard, and it was a difficult game. They make it difficult, the ref makes it difficult. It just had to be a real strong territory game and we had to show lots of character, and Duncan [Weir] knocked the goals."
Weir converted five penalties and registered a drop goal in an 18-point haul to give the Warriors their third away win on the bounce.
And the X-year-old believes the players' work ethic, coupled with the enthusiasm of their supporters, will be a key factor in their opening Pool 3 Heineken Cup clash against Bath on Sunday.
Weir, the RaboDirect PRO12 Man of the Match, said: "There is strength and depth of the squad. There were a lot of changes this week but no-one looked out of place. Everybody is stepping up and making the selection process really hard for the coaches.
"It's nice to be in a winning team, and going into next week it is a massive competition against Bath as they've got a big squad and strength in depth as well.
"(But with) the fans shouting and screaming they mess up the team's line-out call and all that - it makes a massive difference. It gives us a good edge, so having a big crowd is going to be key in a game like that."
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