Alyssa Thompson scored the decisive goal in the 39th minute after the Blues recovered from Arsenal’s strong start.
Chelsea faced further pressure after the break but defended resolutely to secure another narrow victory, having also beaten Austria Vienna 1-0 in midweek.
Bompastor praised her players’ fighting spirit they have worked to develop behind the scenes this season.
“The plan for us this season is mindset and mentality,” Bompastor told Chelsea website.
“I've been talking a lot about being strong and being together. No matter what happens in the game, I always want to see a group of players, and also a team who is together in our high moments when we doing well, but also in the low moments.
“We came into this game well-prepared in terms of knowing we were going to have some moments in the game where we were probably not in control, especially in possession of the ball, because we knew how good Arsenal are at doing that.
“So we prepared the game for those moments. Even if we didn't have the ball, we knew exactly where we wanted to rotate the ball and where we wanted to leave some spaces, and I think we dealt with that really well.”