The north Africans held Joachim Low's men to a goalless draw in 90 minutes and threatened to score themselves on numerous occasions, with Islam Slimani having a goal disallowed for offside in the first half.
Andre Schurrle gave Germany the lead at the start of extra-time and Mesut Ozil thought he had made the game safe on 119 minutes, but substitute Abdelmoumene Djabou still had time to set up a nervy finale.
And after the match Bougherra, who came on during the additional 30 minutes, insisted Algeria could be proud of their spirited display.
"We were very, very close of doing the extraordinary," he told reporters. "We remained level with this great Germany for 90 minutes. We showed great quality and strengths. I'm very proud of what we've done.
"I think the goal in the first half could have made us more confident, there were a lot of chances. But it didn't happen. We reacted well to both goals. When a team plays with such heart, it's the foundation you need to succeed."
Team-mate Rais M'Bolhi earned the Man of the Match award for pulling off a string of impressive saves in the Algerian goal, and after the match he too was upbeat.
"We are part of Algerian football history," he insisted. "We have a lot to build on this. We played on equal conditions with bigger teams than us.
"I have to congratulate my team-mates. There were not many aspects we missed today, even though we conceded two goals."