"We are still precariously close to the relegation zone and with further games against the likes of Cardiff, Huddersfield, Crystal Palace, Newcastle and Fulham to come this side of Christmas, they must discover a ruthless streak to ensure they don't get embroiled in another struggle towards the bottom of the Premier League table".
Chris Wood had seemingly earned Burnley a share of the spoils 13 minutes from time, but the impressive Anderson and substitute Hernandez struck in the final stages to end the visitors' resistance.
Anderson was then denied by the woodwork before putting West Ham back in front with a fight-footed shot into the corner before Javier Hernandez secured the win in stoppage time.
Anderson was the star of the show, however, supporters were lauding the display from Grady Diangana. Instead, it was Burnley who got the game's second goal as Gudmundsson broke the offside trap and finished beyond Lukasz Fabianski after latching on to Ashley Westwood's defence-splitting pass.
The 30-year-old scored eight Premier League goals for West Ham last season and will hope to better that tally this season as the Hammers push for a better league finish under Pellegrini.
Tottenham Hotspur's South Korean attacker Son Heung-min has ended his scoring drought this season after netting two goals against West Ham United.
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I do feel losing the two finals that year was tough on us. We can't have them round here. "It's the wrong place". It is cut-throat - that is the way it is.
Wood's latest goal saw Burnley get back on level terms at 2-2, but the Lancashire club coughed up a soft goal through a defensive blunder to allow Brazilian midfielfder Felipe Anderson to net his second strike of the match.
"A lot of players didn't have a pre-season, that is why it is so hard".
Diangana has had opportunities in the last few games in the West Ham side to impress due to a string of injuries, but he surely now has to be a real contender to become a regular starter under Pellegrini.
Fortunately for Tottenham, their strength in depth shone through as West Ham, who trounced Macclesfield Town 8-0 in the third round, failed to make their chances count.
But attacker Diangana once again stood out from the crowd at the London Stadium and proved that he's around the first team to stay. It will be a tough game, but it's definitely winnable for West Ham. "I don't believe too many players will be ready and fresh to play".