On the stroke of half-time, Cech went to take a goal kick and immediately pulled up.
The 36-year-old signalled to the bench that he had a left hamstring issue and made clear he had to come off.
Cech was given a warm applause by the Emirates crowd and was replaced by Leno.
It’s Leno’s first appearance of the season in the league after matches in the League Cup and Europa League.
Cech could now be out of action for a few weeks but could recover before the imminent international break.
Arsenal face Qarabag on Thursday in the Europa League before a visit to Fulham next Sunday.
League action then takes a break for the resumption of the UEFA Nations League and friendlies.
After those games, Arsenal host Leicester in the league and Cech could be in contention to feature.
The injury will be a blow to Cech after he was given the nod by new head coach Unai Emery over Leno.
Sky Sports pundit Jamie Carragher said the 36-year-old has been helped by the arrival of Leno to provide him serious competition.
He said: “In terms of his performances, I think the signing of the other goalkeeper has actually lifted his performance.
“The saves he’s made, if you think of the game here against West Ham as well, imagine that didn’t go well, or that didn’t go well today, maybe you take one point instead of six points.
“It makes a massive difference.”