On third-and-6, Brees threw 35 yards to a wide-open Marques Colston, who jumped into the air just in front of the goal line at the middle of the field, caught the ball and fell forward into the end zone, making it 16-10 with 12:39 to play.
The Saints (10-3) converted two third downs on their next drive before Brees connected again with Colston on a scoring pass, this time for 28 yards.
The touchdown gave New Orleans a 22-10 lead with 7:01 left, and Colston finished with 105 yards on seven catches.
Brees, who was 36 for 47 for 337 yards, joined Hall of Famer Johnny Unitas as the only quarterbacks to throw a touchdown pass in 40 straight games. He also extended his NFL-record streak of games with 20 or more completions to 33.
He has 39,413 yards passing in 150 career starts. That ranks just behind the 39,498 yards passing Dan Marino achieved in his first 150 starts.
The Titans (7-6) twice had the ball in the final minutes with a chance to win, but fell short.
Faced with another fourth down at the Saints 24 with 2:18 left, Locker kept the ball. Though a replay seemed to show he picked up the needed yard, the officials ruled him down with no gain, and the Titans did not challenge the ruling.
Tennessee held New Orleans on the next drive, and Washington caught a 40-yard pass from Locker with 27 seconds left, setting up another chance for a Titans rally.
Locker spiked the ball at the Saints 5 to stop the clock, but his pass on second down to Marc Mariani in the end zone was batted down by Tracy Porter, and Jo-Lonn Dunbar sacked a scrambling Locker on third down as time expired.
The rookie quarterback entered the game with the score 3-3 with 11:13 left in the second quarter, replacing Matt Hasselbeck.
Hasselbeck hurt his left calf after trying to chase a pass of his that had been tipped and left the field and watched most of the game from the sideline, filling in for Locker on a single play after the backup took a hard hit.
Locker was 13 for 29 for 282 yards and a touchdown, and Washington finished with 130 yards on six catches and a score. Johnson, who was coming off consecutive games with more than 150 yards rushing, had just 23 yards on 11 carries.
Locker ran for a second touchdown, scrambling around the right end and meeting Roman Harper at the corner of the end zone. He stretched the ball over the pylon with his right hand as he spun in the air, and the play was upheld on review to make it 10-9 with 2:19 in the third quarter.
Washington caught a 65-yard TD pass from Locker and broke a tackle before tripping into the end zone to cut New Orleans' lead to 22-17 with 5:58 left.
Penalties and other mistakes kept New Orleans' league-leading offense from getting any rhythm through three quarters, and three times the Saints got within 11 yards of the end zone only to settle for a field goal.
An illegal formation penalty negated an 8-yard pass from Brees to Jimmy Graham that would have made it first-and-goal at the 3 for the Saints. They settled for a 25-yard field goal to make it 3-0 with 9:54 in the first quarter.
Brees drove New Orleans down the field just before halftime, but was sacked by rookie Karl Klug for a loss of 7 yards on first-and-goal at the 5. A well-covered Lance Moore couldn't grab a 11-yard touchdown pass, and John Kasay kicked a 29-yard field goal to make it 6-3 with 15 seconds before halftime.
Brees appeared to connect with Graham on a 4-yard touchdown at the edge of the end zone, but a replay showed the tip of Graham's right foot touched out of bounds. New Orleans coach Sean Payton challenged the ruling, which was upheld, and Kasay kicked a 22-yard field goal to make it 9-3 with 3:51 left in the third quarter.
The Saints also missed another scoring opportunity early in the first half. Darren Sproles returned a punt 82 yards to the end zone, but the score was negated on a holding call on Dunbar.
The only other time the Saints didn't score a touchdown in the first half was in their 31-21 loss at St. Louis on Oct. 30.