Beto and Iliman Ndiaye struck the goals for the hosts in this third round tie.
The victory came just hours before Sean Dyche's dismissal, with Baines and club captain Seamus Coleman taking charge on the night.
The former later said: "It was a unique day in a lot of ways but a nice moment for myself and Seamus and the players as well.
"It being a winning one is definitely a nice moment. The management side is not something I have been striving for but the opportunity arrived today to help out in the situation where the club needed someone and I was more than happy to help out.
"The outcome of winning a game is always a positive one. The lads get to stay in the draw in a cup competition we are all desperate to stay part of.
"The lads played well you could see the commitment. They wanted to dive the stadium and fans something they would enjoy watching. For the most part the performance was positive.
"It has not been that difficult. This is what we do. Players play football and coaches coach. It was just a little different today. It is never nice when a manager loses his job but then everyone had to get on with their jobs as well.
"I am not too sure about that (what is next for the club) right now. Things moved so quickly today. I have definitely not had time to think about what comes next."