Mancini’s men struggled to create any real chances in a tight first-half, aside from David Silva’s 20th-minute effort which struck Thomas Sorensen’s right post.
Republic of Ireland midfielder Glenn Whelan was fortunate to escape unpunished by Howard Webb after the referee missed his two-footed challenge on City midfielder Javi Garcia within minutes of the restart.
Tony Pulis introduced Peter Crouch in a bid to conjure a late winner, but it was the visitors who struck, with the Citizens captain finishing from close range in the 85th minute.
“We won it two years ago and we want to win it again,” said Zabaleta.
He added: “It’s always great to score and three goals this season is not bad for a full-back.
“I need to keep working. It was a massive win on a very tough pitch. It’s always difficult to win here but we were fantastic at the back and aggressive up front.
“It’s so difficult to play Stoke at home. We knew it was going to be tough for us.
“But we were solid from set-pieces.”
With the FA Cup fifth-round draw taking place on Sunday, the Argentina international urged his team-mates to focus on picking up three points away to Queens Park Rangers on Tuesday.
“We’re through to the next round and have to wait for the draw,” he added.
“Now we focus on Tuesday away to Queens Park Rangers. We need to keep winning to catch Manchester United.”