A superb solo score from young wing Kristian Phillips proved decisive at the Stade Yves-du-Manoir as the Scarlets ended a run of six straight defeats in some style.
Man of the match Priestland added 14 points as he missed just once from the tee, with the Wales fly-half slotting four penalties and converting Phillips' 63rd minute try in a 19-13 win.
Easterby said: "It wasn't perfect nor was it really pretty but winning here in France was the most important thing for us.
"We closed out the game very well and we didn't crack at the end when it mattered, like we have done a few times this season.
"The travelling supporters were absolutely outstanding from minute one. That definitely helped the guys out tonight.
"We made life very difficult for ourselves with the back-to-back defeats to Clermont Auvergne last month but we now still might have an opportunity to reach the quarter-finals of the Amlin Challenge Cup."
Priestland, who was in direct opposition with British and Irish Lions playmaker Jonny Sexton in the Pool Four clash just outside Paris, was in impressive form throughout, with his overall control guiding the Scarlets to a second Heineken Cup away win of the season.
But the 27-year-old, who missed out on the chance to challenge Sexton for the Lions through an Achilles injury, is convinced it was his side's character that saw them through a disjointed 80 minutes.
Priestland said: "It was a poor game, to be honest. But we worked hard, even though sometimes we were our worst enemies by putting pressure on ourselves.
"We wanted to simplify our game and our pack was outstanding tonight. We didn't get the accuracy and continuity we expected but we showed great determination.
"We did well overall given that we have been in this situation in a few games this season and it hasn't always turned out well for us. Today, we showed composure and managed to get that win. So that's a good thing for us."
- Sports & Recreation
- Kristian Phillips