The Hammers beat Chelsea and drew at Newcastle United last week, easing their nerves about the drop, but the loss to in-form Burnley left David Moyes's side in 16th place on 31 points, just four points above 17th-placed Aston Villa who have a game in hand.
West Ham's next game is at bottom club Norwich City on Saturday.
"I thought today was must-win, Newcastle was must-win, we have been in that situation for a while, probably since I took over the job," said Moyes, who was frustrated with his team's performance.
The winner came in the 38th minute when Charlie Taylor whipped in a cross from the left and Rodriguez's glancing header flew in off the underside of the crossbar.
Pope, who has now kept 14 clean sheets this season, was in outstanding form, dominating his box and nullifying cross after cross thrown into the penalty area by the home side.
Sean Dyche's Burnley, who are in ninth place on 49 points, had an effort from substitute Chris Wood ruled out in the 76th minute for offside and the Kiwi also forced a good save out of Lukasz Fabianski.
Yet the Clarets had to defend with tenacity in the final period as West Ham piled bodies into the box -- only to come up against the inspired Pope, who caught or punched clear every danger.
Substitute Sebastien Haller had the best chance, moments after being introduced, but his poked effort was kept out by the outstretched leg of Pope.
"The two centre halves were outstanding, the keeper was outstanding," said Dyche.
"They asked a lot of questions, had a couple of golden moments, and our keeper was there when needed," he added.
Burnley have lost once in their last 12 Premier League matches and won three of their last four games.