The Magpies needed to beat a Southampton side weakened by injury to stand any chance of qualification.
They had enough chances in a pulsating match to have snatched the win to keep alive their hopes but the Saints never gave up.
In a match of a multitude of chances, Southampton goalkeeper Alan Blayney made three outstanding saves to deny both Ambrose and Steve Caldwell and both players struck the bar in the second half.
Newcastle were ahead after seven minutes when the powerful Shola Ameobi held off a challenge by Claus Lundekvam and gave Blayney no chance with a low shot.
But Southampton equalised after 20 minutes when Shay Given failed to hold a shot from Anders Svensson and the rebound fell into the path of James Beattie.
Both sides then went close to taking the lead, Kieron Dyer and Gary Speed for Newcastle and Yoann Folly and Beattie again for Southampton.
But after 36 minutes Newcastle's lead was restored when Ameobi broke down the left and Lee Bowyer clinically put away his centre.
Two minutes later Southampton were level again when a low centre from Fabrice Fernandes was deflected into his own net by Titus Bramble.
Newcastle dominated the second half but Blayney, making his debut in place of the ill Antti Niemi, made an outstanding block to deny Alan Shearer and then made equally important saves to keep out Ameobi and Shearer again.
Southampton lost Lundekvam with a back injury in the first half and when substitute Kevin Phillips came off with four minutes remaining, Saints were reduced to ten men.
But with two minutes remaining, his absence appeared to matter little as another substitute, Leandre Griffit, ran on to a headed pass from Beattie to shoot low past Given.
Deep into stoppage time, with Newcastle committed to all-out attack, Ambrose found room in a packed area to score with a low right-foot shot but it proved to be the last kick of the match and as far as Champions League qualification was concerned, too little too late.