Premier League strugglers Cardiff City and Huddersfield Town both crashed out of the FA Cup to lower league opposition on Saturday.
Burnley have three England goalkeepers on their books, but manager Sean Dyche does not expect any of them to leave in January.
Huddersfield Town are eight points from safety in the Premier League but boss David Wagner remained hopeful after losing 2-1 to Burnley.
A red card for Christopher Schindler handed Burnley the momentum before they ran out 2-1 winners away at Huddersfield Town.
Tom Heaton was preferred to Joe Hart for the first time in the Premier League and Sean Dyche's call was vindicated as Burnley beat West Ham.
Burnley boss Sean Dyche was left "super disappointed" with his team's terrible first 30 minutes against Everton on Wednesday.
Everton scored three times in the opening 22 minutes on the way to a 5-1 crushing of Burnley at Turf Moor.
Unai Emery had to replace Nacho Monreal in the first half of Arsenal's win over Burnley as his defensive injury problems deepened.
Unai Emery disagreed with Sean Dyche after the Burnley boss accused Arsenal players of diving during their 3-1 victory.
Ashley Barnes scored a consolation for Burnley, but they failed to mount a comeback as Arsenal ran out 3-1 winners at Emirates Stadium.