The Czechs entered the match in second place in the group, five points behind leaders Germany who are on 22 and three ahead of the Irish after eight matches.
With six points now between themselves and Ireland, the Czechs need just one win from their final three games against Germany, Cyprus and Slovakia.
"We still have important games ahead, we can't say we have advanced. But this was a big win for us, it helps separate us from the others," said Czech coach Karel Brueckner.
The Czechs pounced on a tentative Irish team from the kickoff and Jankulovski struck in the 15th minute when he combined well with Jaroslav Plasil.
The AC Milan defender took a return pass and then coolly slotted home a low shot into the Irish goal.
"We showed a lot of heart and I think this performance should quieten some of our critics. The goal was a great example of how our team work is developing. We're now closer to our dream," Jankulovski said.
Tempers flared in a match that that at times looked like getting out of the control of Greek referee Kyros Vassaras.
Stephen Hunt, an early first half substitute for Ireland, was jeered loudly by the crowd, recalling his reckless challenge in an English Premier League match last year that put Czech goalkeeper Petr Cech in hospital with a severe head injury.
The boos reached their worst in the 61st minute when his aggressive tackle from behind on Jan Polak drew an immediate red card from Vassaras, who also handed out nine yellows during the match.
Even though they were down to 10 men, Ireland still mounted some dangerous counter-attacks and almost capitalised on a Cech mistake in the 71st minute, but defender David Rozehnal cleared Paul McShane's weak shot just before it crossed the line.
"We are greatly disappointed. We started badly, but we got back to our top form. Even when we went down to 10 men, we still had chances," said Irish coach Steve Staunton.
"It's out of our hands, it's now up to the Czechs," he added, referring to Ireland's chances of overtaking the Czechs.