The Hammers were in desperate need of three points ahead of their clash with Sean Dyche's side, having just pulled themselves out of the relegation zone with Sunday's 2-2 draw with Liverpool.
Bilic's side dominated the first period, and were rewarded with Mark Noble turning home the only goal of the game in the first-half stoppage time. That win has pushed them four points above the relegation zone, and Bilic praised his side's character in their first league win since October.
“It was an important win for us," Bilic told whufc.com. "Considering the situation we were in and the fact this game was being talked about for a long time."
“To be fair, it [felt like it] was a long game," he stated. "One of the longest that I’ve been involved in but it wasn’t for the players."
We got three points we showed the character and all that," the coach admitted. "I think we deserved it for the way we played in the first half when we were absolutely dominant."
“That cramp, I’d call it, that we are feeling at the moment – that is quite normal considering the situation we’re in – should go," Bilic added. "The only way it goes is if you are moving up the table."
Could have done better
The three points didn't come easy however, as they were battered by the clarets in the second period. The visitor had a handful of excellent chances from Sam Vokes and Scott Arfield, and Bilic admitted that his side could have done better.
"We are afraid that we might concede a goal," Bilic told Sky Sports. "That is quite normal in the position that we are when you are desperate for points."
We conceded late against Liverpool," the coach stated. "So it is going to affect us, but not like that [against Burnley]."
"OK they were throwing the players in there with long balls and all that," the Croatian admitted. "But we should have done better in the second half."
"Those three points should give us more confidence and we won't have another second half like that," Bilic concluded. "After the first half, especially when you score a goal just before half time, it shouldn't happen."
West Ham United will take on Hull City at the London Stadium on Saturday, December 17 with kick-off at 3pm GMT.