Arsenal entered a new era last weekend as they played their first Premier League game since 1996 without Arsene Wenger at the helm.
But Unai Emery tasted defeat on his domestic bow as goals from Raheem Sterling and Bernardo Silva handed Manchester City a 2-0 win.
Arsenal failed to get their first Premier League points of the season on the board this weekend either.
The Gunners were beaten 3-2 at Chelsea having initially fought back from going two goals down inside the first 20 minutes.
Crooks, who spent five years with Arsenal’s north London rivals Tottenham, says Emery’s men warranted a point for their performance against Chelsea.
But he has predicted they could miss out on securing a European berth come the end of the season.
“Last week, against Manchester City, I said Arsenal would do well to finish in the top six,” Crooks told BBC Sport.
“This performance against Chelsea was much improved.
“In fact, they deserved a draw.
“But I’ve not changed my position.”
Crooks bemoaned Arsenal’s lack of prowess in front of goal against Chelsea, where Pierre-Emerick Aubameyang missed two glaring opportunities.
“This is a very different Chelsea under Maurizio Sarri than it was under Antonio Conte,” Crooks said.
“For a start, the Blues played parts of the game as though they had forgotten how to defend.
“It’s not that long ago that Chelsea had the best defensive record in the league and, with better finishing from Arsenal, the Gunners could have gone on to win this London derby.”
Arsenal will be eager to get off the mark when they return to Premier League action this weekend.
On Saturday, Arsenal play host to West Ham, who snapped up Jack Wilshere on a free transfer this summer.