Midfielder Phillips' sixth goal of the season gave the Baggies a sixth-minute lead and Robson-Kanu's second-half penalty ensured they comfortably avoided an upset at the bet365 Stadium and leap-frogged Preston at the summit.
Scotland international Phillips was never going to miss from close-range for his second goal in as many games as Slaven Bilic's promotion-hungry side stretched their unbeaten run to five games.
And Robson-Kanu's third goal in four matches inflicted an 11th defeat in 15 games on Stoke who, only three days ago, sacked manager Nathan Jones after a miserable start to the season. They are now six points adrift of safety.
Victory puts the Baggies two points ahead of Preston, Leeds and Swansea.
But it was the visitors who took the lead just two minutes later when Phillips slotted home with a sweet strike into the bottom left corner.
West Ham loan winger Diangana worked a clever one-two with skipper Jake Livermore and delivered across the face of goal where the unmarked Phillips easily slotted home.
Defender Darnell Furlong tried his luck from distance, but his right-foot strike flew well wide for the visitors before West Brom striker Charlie Austin ended up with a bloodied nose after a stray arm from defender Liam Lindsay just before half-time.
Carter-Vickers should have levelled for the Potters straight after the break, but the right-back somehow blazed over from point-blank range.
Senegal midfielder Badou Ndiaye blazed over from distance with a very optimistic long-range effort as Stoke pushed to restored parity.
But no matter how hard they tried, the hosts struggled to create any clear-cut opportunities for their frustrated fans.
Carter-Vickers' reckless challenge on Diangana resulted in West Brom's penalty – and Robson-Kanu made no mistake to seal the points.
The Welshman had only been on the pitch for six minutes, but he calmly slotted into the bottom left corner as keeper Jack Butland dived the opposite way.
And it should have been 3-0 moments later, but this time Robson-Kanu headed Phillips' inch-perfect across against the right post.
Robson-Kanu also forced a save out of Butland at the death as West Brom deservedly picked up maximum points on their travels.