The Bees won 3-1, with all four goals coming in the final ten minutes as Igor Thiago's brace proved decisive.
"It was a well deserved win, we had to earn it," said Andrews. "You don't get given anything in this league; every result is hard-earned.
"Naturally, when the big boys come to town, there's a feel around the club, and the feeling of getting something against a Liverpool or a Newcastle is, 'Oh that's an amazing result', which it is.
"But, today, Burnley are fighting for their lives. They're a well-drilled team with a very good coach (Scott Parker) - they're doing everything they can to stay in the division.
"It was always going to be a game we had to stay patient in, be resilient, do the basics well, keep knocking at the door, and, in the end, we were able to get the win."