Baker came from a set down to defeat Frenchman Teixeira 5-7 6-2 6-1 6-2 and take his place in the main draw.
He was a Wimbledon junior quarter-finalist and French Open junior finalist in 2003, but was out of the game completely for seven years after suffering a succession of injuries. He went through five operations - including three on his hips - before returning to the professional game last summer. Since then he has steadily climbed the ATP rankings and is currently 124th in the world.
“It feels great and is very exciting,” Baker told the official Wimbledon website.
“I started off a little slow today but I was able to turn it round pretty quickly. Once I got the first breakpoint the floodgates kind of opened up and I was able to break a lot. I had two match points. It’s never easy closing it out.”
Baker's injury was originally sustained at Wimbledon in 2005, when he slipped badly and tore his medial collateral ligament in a qualifying match against a young up-and-comer named Novak Djokovic
But despite that injury, his hip problems (which had plagued him since he was a boy) and subsequent elbow and hernia woes, he said recently that he never considered giving up his battle to play at the highest level.
"I never reached the point where I said I was ready to throw in the towel," Baker said.
"You have to be realistic. I wasn't going to keep on having major surgeries to try to continue my career. Fortunately I have been able to start feeling better as of 2011 and here I am now."
Baker was one of 16 men to win through on day three at Roehampton.
Thursday sees the final day of women's singles qualifying.
Men's singles qualifiers:
Jurgen Zopp (EST)
Adrian Menendez-MacEiras (ESP)
Guillaume Rufin (FRA)
Michael Russell (USA)
Jesse Levine (USA)
Florent Serra (FRA)
Ryan Sweeting (USA)
Dustin Brown (GER)
Simone Bolelli (ESO)
Jimmy Wang (TPE)
Brian Baker (USA)
Kenny De Schepper (FRA)
Ruben Bemelmans (BEL)
Inigo Cervantes (ESP)
Jerzy Janowicz (POL)
Andrey Kuznetsov (RUS)