West Ham United 0, Burnley 1 FT
A first half header from Jay Rodriguez, who did brilliantly to guide a Charlie Taylor cross in off the bar, proved enough, and moved the Clarets within four points of Wolves in sixth - who come to Turf Moor next Wednesday.
Dyche, unsurprisingly, put out an unchanged side from the 1-1 draw with Sheffield United on Sunday, with Anthony Glennon replacing Bobby Thomas on the bench.
David Moyes replaced Manuel Lanzini with Andriy Yarmolenko, with the luxury of £45m record signing Sebastien Haller back on the bench, for his first involvement since the restart.
Having won three of their last five away games - as many as in the previous 16 - the Clarets were out to build on their win at Crystal Palace last Monday, one of only four victories in 31 visits to the capital.
Moyes, meanwhile, was looking to get one over on Dyche for the first time, having drawn two and lost three against the Clarets boss.
Czech Republic international Tomas Soucek had netted in the previous two games, and was looking for another from the off, finding the Sir Trevor Brooking Stand from distance early on, before heading Aaron Cresswell's cross well over.
But in an open first half, Burnley also posed a threat, and Erik Pieters, again impressing in midfield, this time on the right, should have done better than lift his shot over after Matej Vydra held the ball up well from a Charlie Taylor centre.
Vydra headed another great chance at Lukasz Fabianski from a Dwight McNeil free kick, as Burnley enjoyed a good spell of possession.
After the drinks break midway through the half, the Hammers came on strong, as Soucek forced a fine save from Pope from a flying volley from Ryan Fredericks' cross - firing straight at the keeper when either side would likely have produced a goal.
Moments later, Pope was worked again by Michail Antonio, getting down to his near post to save well.
Andriy Yarmolenko then miscontrolled in front of goal after Antonio beat Kevin Long and fed the ball across the area.
But with the Hammers enjoying their best spell, Burnley produced a quite brilliant goal, as Taylor got down the left, and from his precise cross, Jay Rodriguez got across Cresswell and guided a header in off the bar - a brave finish with Cresswell's foot high.
Still the chances came at the other end, as Antonio shaped an effort wide when gifted with a clear run at goal after Ashley Westwood lost the ball.
Again Burnley responded, and Vydra dragged a volley wide when well placed from Pieters' lovely lofted pass.
Pope had to get down well to save at his near post again from Yarmolenko before half-time, but the Clarets went in ahead, eyeing another big three points.
Burnley started the second half on the front foot, with a burst from Vydra ultimately coming to nothing, before the Czech striker was denied at near post from a McNeil cross after a clever Rodriguez back heel.
Just before the hour, James Tarkowski was the first man in the book, for a foul on Jarrod Bowen, as Wood came on for Vydra for his 100th Premier League appearance, looking to add to his six goals in five games against the Hammers.
Haller replaced Yarmolenko shortly after, and his first involvement was to squander a huge chance, after Antonio forced the ball goalwards, the Frenchman snatching at his effort, with Pope saving with his feet.
Burnley had chance to release the pressure valve, but Taylor was brought down on halfway by a cynical foul from Fredericks, who was given a yellow card.
Wood came close to stretching his superb strike rate against the Hammers, when his improvised flick forced a save from Fabianski,
He had the ball in the net with 15 minutes remaining, but a VAR check ruled it out for offside, after a lovely move saw Brownhill tee up Bardsley, whose low cross was turned in by Wood.
Wood was again denied moments later, forcing a good stop from Fabianski after racing onto Taylor's pass and driving at Issa Diop.
Burnley were far from sitting on their lead, and Westwood screwed a shot wide inside the last 10 minutes, but their resulte defending saw them through six added minutes to claim a second win in London in nine days.
West Ham United: Fabianski; Fredericks, Diop, Ogbonna, Cresswell; Soucek, Rice; Bowen, Fornals (Ajeti 86), Yarmolenko (Haller 62); Antonio.
Subs: Randolph, Balbuena, Lanzini, Noble, Wilshere, Masuaku, Johnson.
Burnley: Pope; Bardsley, Long, Tarkowski, Taylor; McNeil, Brownhill, Westwood, Pieters; Rodriguez, Vydra (Wood 58).
Subs: Peacock-Farrell, Gudmundsson, Brady, Thompson, Dunne, Benson, Goodridge, Driscoll-Glennon.