Former Celtic boss manager, who took over from Gareth Southgate a fortnight ago, saw his new side fire blanks as they crashed to a second straight defeat.
Boro almost took the lead after quarter of an hour when Leroy Lita's deflected shot forced a sharp save from Julian Speroni.
Marcus 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.
And Boro were made to pay in the 65th minute when Palace broke the deadlock in style. From the visitors' corner, Shaun Derry fed 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.