The Blues have won only two of their last seven matches in all competitions in front of their own fans and have lost the last two since new year.
Cech said: "This was a great win as this is a big weekend for us.
"But it was vital to get the win and close the gap to put pressure on the other teams.
"Most of them play Sunday but Tottenham, who are in the pack for a top-four finish, drew with QPR so we knew if we won our game it would give us a chance of closing the gap.
"If we can get three wins from three the table will look different for us.
"This was a great win - Stoke's home record was brilliant and they are very strong at home. But now if we don't confirm it by winning at home then you waste it. You don't want to do that. We want to push up the table."