After Sanchez was gifted an opener by Steve Cook, Callum Wilson had equalized for the Cherries from the spot. But second half goals from Theo Walcott and Sanchez himself saw Eddie Howe's men get undone by an Arsenal that showed flair.
A dangerous challenge
Sanchez, whose strike in stoppage time had put the game out of Bournemouth's reach, had tackled Arter from behind in the 62nd minute to send the Irishman down and left him crying out in agony.
Hackett wrote in his Daily Telegraph column about why and how Sanchez didn't deserve to be on the pitch following his lunge on Arter and also told that referee Mike Jones and his assistants got three or four decisions wrong.
"They are exactly the sort of tackles that we do not want in the game and must be dealt with firmly," said Hackett, going onto to admitting that it "should have been a red card."
Hackett also labelled the tackle as being 'dangerous' and felt that Howe is right to "expect the officials to have spotted" the challenge on the Bournemouth midfielder.
The 72-year-old also told in his column that the penalty given to Bournemouth was 'soft' and the coming together between Wilson and Arsenal full back Nacho Monreal "did not warrant infringement."'
Liverpool tie awaits Howe
The defeat to Arsenal, which saw Bournemouth come up with a spirited showing, isn't the end of what promises a tough run of game for Howe's men. A home game against Jurgen Klopp's Liverpool awaits them on Sunday.
The 3-1 defeat sees Bournemouth lose their third game in five outings and makes sure that they slip to the 12th position in the table.