The Hammers defended resolutely for the third Premier League game in a row, stifling Arsene Wenger's star-studded side with their organisation, teamwork and never-say-die attitude.
Ogbonna led the way, making a team-high eleven clearances and two interceptions, before Chicharito came within inches of snatching all three points by rattling the underside of the crossbar with seconds remaining.
"We are not individuals. If I play well, it's because the team played well. I think the team is playing so much better than before and we have to keep this up until we are dead!
"We need to keep the concentration in every detail, because the details can change. If Chicha's shot had gone in, it would have been totally different. When they got the free-kick right at the end with [Mesut] Ozil, it could also go in, but it hit the wall.
"In football, detail is so important so we have to be focused on everything."
I think the team is playing so much better than before and we have to keep this up until we are dead!
Wednesday's stalemate made it two clean sheets in succession and five for the season.
Ogbonna said the team's improved defensive output is testament to the work the players are putting in and the methods of new manager Moyes and his staff.
"I'm really proud of the team because we kept a clean sheet and we did really well. Hopefully, maybe on Saturday we can get three points.
"It's not only about the clean sheets, but the way we have approached the last three games was so different than the way we approached the other games.
"In this game, we wanted to get something important and for us, for the staff and for everybody, it's really good."
Next up for the Hammers is a trip to Stoke City on Saturday, where Ogbonna believes West Ham have an opportunity to extend their unbeaten run to three matches with a first away win of the season.
"Everybody wants to show what they can do. The manager has put some basics in place which have helped us to do that.
"It's difficult at Stoke, but this is a big chance to get three points up there. We needed to change attitudes and the way we approach games is much better than before."