Chelsea were denied a first win under Frank Lampard after Leicester City fought back to draw 1-1 on Sunday, while Sheffield United recorded a hard-fought 1-0 victory over Crystal Palace.
Wilfred Ndidi cancelled out Mason Mount's early opener at Stamford Bridge to leave Chelsea with a solitary point in the early stages of Lampard's regime.
Leicester looked the side more likely to win the game after equalising in the 67th minute, but a couple of good chances went begging and they had to settle for a second successive draw.
Like Chelsea, Palace have also made a slow start to the new campaign after going down to a John Lundstram goal at Bramall Lane in the early kick-off - the Blades securing their first top-flight win in more than 12 years.
REDEMPTION FOR NDIDI
Lampard's first taste of Premier League management ended in a heavy 4-0 loss to Manchester United last weekend, followed up by a penalty shoot-out defeat to Liverpool in the UEFA Super Cup, and his home bow also ended in disappointment.
Chelsea got off to a positive start as Mount was rewarded for pressuring Ndidi high up the field, the English youngster taking the ball away from his opponent and powering a shot past Kasper Schmeichel for his first goal in Chelsea colours.
But Ndidi went on to atone for his error by converting the impressive James Maddison's corner, heading his team level.
Perhaps tired following their midweek exertions in Istanbul, Lampard's side were left holding on in the final quarter of the match, Maddison firing over from a good position and Jamie Vardy also passing up a presentable opportunity.
LUNDSTRAM FIRES SHEFFIELD UNITED PAST PALACE
Promoted Sheffield United marked their first Premier League game at Bramall Lane since 2007 with all three points thanks to Lundstrum's 47th-minute strike.
The 25-year-old fired home from 10 yards after Luke Freeman had been denied by Vicente Guaita, in the process joining the list of players to have scored in the top four divisions in England.
Despite having Wilfried Zaha in their starting XI, Palace struggled for attacking inspiration following last week's stalemate with Everton.
United, by comparison, are seventh in the early standings after adding to the point they claimed at Bournemouth.