Late winners ensured they didn't prove costly.
Haaland struck a first-half penalty against the post and squandered a slew of other chances in an unusually profligate display from the leading scorer in the league last season.
He finally found the net — doesn't he always? — in the 63rd to put City ahead but then it was Walker's time to mess up, when he attempted a backheel in his own area and gave up possession from which Sheffield United equalized.
Rodri, the scorer of City's winner in the Champions League final in June, came up with another timely goal to continue the champions' 100% start to their title defense.
Liverpool left it even later, with substitute Darwin Nunez scoring the second of his goals in the third minute of stoppage time to earn a 2-1 win at Newcastle that came in spite of a woeful defensive display by the Reds in the first half at St. James' Park.
After getting booked in the sixth minute and then escaping a sending-off for a cynical foul soon after, Alexander-Arnold completed a hat trick of mistakes by letting Mohamed Salah's backward pass slip under his boot. That set free Anthony Gordon unchallenged and he slotted a finish between Alisson Becker's legs in the 25th.
Then, three minutes later, the usually unflappable Van Dijk made a rash tackle — as the last man — on Alexander Isak that earned him a straight red card, seemingly ending any realistic hopes of Liverpool avoiding defeat.
However, Nunez took advantage of more uncertain defending, this time from Sven Botman as he failed to deal with a through-ball, to equalize in the 81st.
The Uruguay striker then ran onto a pass from Salah to convert an almost-replica finish and secure an unlikely triumph.
Newcastle, which will play in the Champions League this season, has only won one of its first three games, though the Saudi-controlled northeast team has had a tough start against Aston Villa, Man City and Liverpool.
Villa was thrashed 5-1 by Newcastle in the opening round but has bounced back with two wins, the latest coming at Burnley 3-1 on Sunday.
Matty Cash, a right back, scored two of Villa's goals.
Steve Douglas is at https://twitter.com/sdouglas80
AP soccer: https://apnews.com/hub/soccer
Steve Douglas, The Associated Press