Stuttgart were dominant in the first half, locking Rangers in their own half, but were punished for stepping off the gas in the second, with Algeria international Bougherra's strike replying to Russian Pavel Pogrebnyak's first-half goal.
"We were not acting, only reacting," said Stuttgart coach Markus Babbel. "We let it slip out of our hands, we only ran behind them and did not go strongly into the challenges in the second half. These mistakes are punished."
Pogrebnyak slotted home to give Stuttgart the lead on 18 minutes, after Brazilian-born Cacau took advantage of a deflected Sasha Papac clearance to race forward and feed the striker inside the box.
Stuttgart dominated possession and could have added another through Cacau, a Germany international, when he fired into the side netting.
Rangers, who had failed to create a single scoring chance in the first half, came out more aggressive in the second but it was Cacau who missed the best early chance when he struck from 18 metres only to see his shot sail over the bar.
The hosts thought they had done enough to win, gradually surrendering valuable space in midfield and allowing the Scots to become increasingly threatening.
Stuttgart, in the same group as Rangers on both of their previous Champions League appearances, were punished for their passive second-half play when Bougherra combined well with Jerome Rothen to fire home the equaliser in the 77th minute.
Rangers turned up the heat after the goal and could have won in the final minutes. Steven Naismith's shot was blocked by keeper Jens Lehmann and Steven Davis hit the post, with the German keeper well beaten, a minute from time.
