Pulis urges calm after West Brom progression
Saido Berahino's brace, along with further goals from James Morrison and Saido Berahino, helped the hosts to a dominant win at The Hawthorns, Morgan Amalfitano's late dismissal compounding matters for West Ham.
With West Brom still in danger of relegation from the Premier League, Pulis urged his side not to get carried away by reaching the quarter-finals.
"We mustn't get carried away, it was a good performance and we're pleased but there's a long way to go in the league and we need a little bit of luck with the draw," he told BT Sport.
"I thought we were very compact, with and without the ball, so we made it very difficult for West Ham. Things broke for us, things went our way.
"Sometimes you need that bit of luck and I'm really pleased for the players and the fact we've scored four goals."
Pulis also reserved praise for two-goal hero Ideye, who was reportedly set to exit the club in the January transfer window.
"I don't honestly believe there was a deal agreed, they were talking but there wasn't a deal agreed," the Welshman added.
"All the speculation about him going was speculation and even if we'd have brought Carlton [Cole] in, we'd still have needed another forward to have stayed.
"I'm not so sure he would have left, but he's done absolutely brilliantly for us. There was a weight on his shoulders and scoring goals removes that weight off.
"He's blossomed and he looks a good player."