Pep Guardiola's City had been frustrated by Everton goalkeeper Maarten Stekelenberg, who saved spot-kicks from Kevin De Bruyne and substitute Sergio Aguero during proceedings. When you are playing in the Premier League then it is not that high.
The Spaniard rose to head home David Silva's cross with 18 minutes to go after Everton had taken the lead at the Etihad Stadium through Romelu Lukaku. Less of a threat in the second period and was withdrawn for the final quarter of the game. Eyebrows were raised when Alonso signed for the Blues, but the 25-year old Spaniard has adapted remarkably well to his left wing-back role and will nail down a regular first team berth if his good form continues.
Swansea were offered a lifeline five minutes later when an error by Switzerland worldwide Granit Xhaka gifted possession to Gylfi Sigurdsson, who struck a attractive curling shot beyond Petr Cech and into the far corner of the net.
Mesut Ozil was also on target for Arsenal, with Gylfi Sigurdsson and Borja netting for Swansea City.
Leicester has sustained three-goal defeats before each of its games in the Champions League group stage this season, having already lost 4-1 to both Liverpool and Manchester United.
Against the run of play, Ryan Mason equalised for Hull after 34 minutes but Cook quickly headed Bournemouth back in front.
Joe Allen proved Stoke City's hero with a first-half double setting the Potters on the way to a 2-0 victory over Sunderland in the dogfight between the Premier League's two bottom clubs.