Saunders had made six previous starts in Texas and had not won any of them, but Baltimore Manager Buck Showalter chose him for the winner-take-all contest because of his experience.
He worked his way out of trouble spots in the first and fourth innings before the Baltimore bull pen wrapped up the victory in the Orioles' first playoff game in 15 years.
Baltimore will now take on the New York Yankees in the best-of-five division round beginning Sunday.
The loss ended a late-season Texas collapse that saw the Rangers lose a four-game division lead in the last six days of the regular season. Oakland swept the Rangers in a season-ending series to finish one game ahead of Texas in the American League West.
After reaching the World Series two years in a row, the normally productive Texas lineup faded at the end of the campaign. That trend continued with Saunders on the mound Friday.
The Rangers put runners on first and third with nobody out in the first inning, but Josh Hamilton grounded into a double play to prevent the chance of a big inning.
Texas then put runners on first and third with one out in the fourth with the score tied at 1-1, but Saunders struck out Mike Napoli and got Geovany Soto to ground into a fielder's choice to end the threat.
Saunders won a post-season game for the first time in five tries, having suffered one loss in four previous playoff starts with the Los Angeles Angels and Arizona.
Baltimore scored a run in the first on J.J. Hardy's RBI single and took the lead for good in the sixth when Hardy scored on a sacrifice fly by Adam Jones. A single by Nate McLouth in the seventh brought home the third Baltimore run and the Orioles tacked on two more in the ninth on a run-scoring single from Manny Machado and McLouth's sacrifice fly.
Darren O'Day followed Saunders to the mound and gave up a hit over two innings.
Johnson had 51 saves during the regular season, three more than second-place Fernando Rodney of Tampa Bay and nine more than anybody else.
The game brought an end to the first round of wild-card playoff games in baseball history and turned out to be much less dramatic than the National League version played earlier in the day.
St. Louis advanced in that one over Atlanta 6-3 in a game played under protest by the Braves after a controversial infield-fly rule call sparked an 18-minute delay brought on by debris thrown onto the field by Atlanta fans.