Bent and strike partner Leroy Lita both fluffed their lines in front of goal and Ambrose showed the visitors how it should be done when he applied the finishing touch to a sweeping Palace counter-attack midway through the second half.
The midfielder now has 10 goals to his name since arriving on a free transfer from Charlton, is surely one of the bargains of the summer.
He nearly opened the scoring in the second minute with a curling free-kick which flew inches wide.
Boro almost took the lead after quarter of an hour when Lita's deflected shot forced a sharp save from Julian Speroni.
Bent was guilty of a shocking miss five minutes later, Adam Johnson cutting the ball back to the on-loan Birmingham striker who was right in front of goal but shanked his effort horribly wide.
Palace were faring little better, however, with Alan Lee heading narrowly wide and Johnny Ertl firing over the crossbar.
The lively Freddie Sears should have got a clear run on goal just before half-time but was halted by a deliberate handball from Osbourne, who escaped with a yellow card.
Strachan had his head in his hands after the interval when a swift Boro break left lone Palace defender Danny Butterfield outnumbered four to one.
But Lita inexplicably opted not to square the ball as his three unmarked team-mates pleaded for a pass, and having been forced wide by Butterfield shot harmlessly across goal.
And Boro were made to pay in the 65th minute when Palace broke the deadlock in style.
From the visitors' corner, Shaun Derry Alan Lee who surged forward and swung in a diagonal cross which Neil Danns nodded on to Ambrose for a simple sidefooted finish.
Gary O'Neil should have salvaged a point for Boro as he broke clear but Speroni pulled off a fine save and the Argentinian also blocked Jonathan Franks' late shot to secure Palace's victory and leave Strachan plenty to ponder.