A goal from Jay Rodriguez put the Clarets ahead in the first-half but Sean Dyche was indebted to his goalkeeper for keeping the hosts at bay.
Andriy Yarmolenko came in for Manuel Lanzini but, other than that, David Moyes' team were unchanged from their win against Chelsea.
Dyche also went with the same starters despite the return of Chris Wood, Robbie Brady and Johann Berg Gudmundsson.
Burnley started the game well and Erik Pieters could have given them the lead in that period. The left-back, currently playing at right-midfield, found himself in the opposition box but lashed the ball over after Matej Vydra teed him up.
Pope's reflexes were called upon twice in quick succession. He first got his body in the way of Tomas Soucek's powerful volley before flicking out a leg to save Michail Antonio's driven shot.
Antonio was West Ham's go-to man and it was the auxiliary striker's time to turn provider. The 30-year-old skipped past Phil Bardsley and Kevin Long before teeing up Yarmolenko. The Ukrainian looked set to slam the ball home but Charlie Taylor recovered with a superb block. Burnley survived.
Against the run of play, though, the Clarets took the lead.
Taylor surged forward and played an inviting ball into Jay Rodriguez and the striker guided a brilliant header in off the cross-bar. It was a goal reminiscent of his winner against Watford.
West Ham nearly responded immediately as Antonio surged past and sleeping Long but he rattled the outside of the post with only Pope to beat.
The Hammers continued to pile on the pressure and Antonio forced Pope into a smart save at his near post.
Moyes would have been frustrated by the half-time score but his team struggled to keep their momentum after the break.
It was a much more tepid affair with Burnley happy co contain and West Ham struggling to get going. That prompted Moyes to fast-track club-record signing Sebastien Haller's return from injury and the substitute nearly made an instant impact. Pablo Fornals was taken off and a 4-5-1 become a 4-4-2.
The striker had been on the field for less than a minute and found himself in on the goalkeeper with the Burnley defence slow to react to West Ham's shift in formation. Haller looked set to equalise but was denied by Pope's outstretched foot.
Wood, off the bench for Vydra before the hour, thought he had sealed the match with a poacher's finish from Bardsley's low cross but the New Zealand international was narrowly offside.
Wood had another opportunity shortly after, tricking his way inside Issa Diop but Fabianski was equal to his placed shot.
West Ham United: Fabiabski 6; Fredericks 7, Ogbonna 6, Diop 6, Cresswell 6; Rice 6, Soucek 7; Yaramelenko 5, Fornals 5, Bowen 6; Antonio 8.
Subs used: Haller 5, Ajeti n/a.
Burnley: Pope 9; Bardsley 7, Tarkowski 8, Long 6, Taylor 8; Pieters 7, Brownhill 8, Westwood 7, McNeil 7; Rodriguez 7, Vydra 6.
Subs used: Wood 6.
Nick Pope: Burnley's stopper is leading the race for the Golden Glove and he certainly earned his money tonight. He made several fine saves and commanded his box excellently. You might argue West Ham's finishing was poor but Pope used his body very well to execute blocks.
Michail Antonio: He was on a mission in the first-half and terrorised that Burnley defence. Antonio might not offer guaranteed goals but he is a nuisance and a livewire. His influence waned after the interval.
There wasn't much wrong with this West Ham performance. They created a number of opportunities against a stingy team like Burnley but there was little purchase.
Antonio is a real enigma of a player but he is capable of delivering the necessary goods. He has played in defence, midfield and attack and that versatility is symptomatic of his brilliant attitude. He works hard and makes a nuisance of himself. The end product is either wonderful or woeful and that can be infuriating at times. But his enthusiasm and determination is exactly what West Ham need right now.
Moyes has a squad that is filled full of quality but it also has some honest characters in the mix. They lost this match against Burnley because they lacked a cutting edge in the final third. Most importantly, the attitude was good and the plausibility of getting a result was on show. That's not something you can guarantee with other teams near the bottom.
Pope's rise at Burnley has been phenomenal. Three years ago, he hadn't played Premier League football and now he might win the Golden Glove! He owes a lot to Dyche's emphasis on structure. He has a strong defence in-front of him and a team committed to grinding out the points. But he also deserves a lot of credit for his own performances. You will struggle to find many goalkeepers who command their box better.
He is England's best overall goalkeeper but his distribution is the niggling factor. Jordan Pickford and Dean Henderson are significantly better with the ball at their feet and, at that level, it become a big factor. Nonetheless, he is a sensible presence between the sticks and his big body frame is tough to beat.
Since the restart, Henderson's performances have been patchy and Pickford has looked a little shaky in certain moments. Pope hasn't been faultless. But this performance will back up those who support his right to wear the Three Lions jersey.
The Hammers will be hoping to bounce back at rock-bottom Norwich City on Saturday.
Burnley have only lost once since the restart but that record will be tested against Liverpool, the Champions.