Charlie Austin converted from the spot late in stoppage time to sink a spirited 10-man effort from West Ham.
After knocking at Crystal Palace's door all game, Sadio Mane finally found a way through at Anfield.
Jese scored the lone goal of the game on his Stoke City debut to help the Potters hold off Arsenal at home.
Hal Robson-Kanu's lone goal of the game outweighed his late red card as West Brom held on for the narrow win at Burnley.
Leicester blanked Premier League newcomers 2-0 courtesy of goals from Shinji Okazaki and Harry Maguire.
Saiz recorded a goal and an assist for Leeds in a 2-0 win over Sunderland to remain unbeaten after four matches.