With Tulane facing a third-and-9 against Houston with 2:31 left in Saturday's game, it would have been easy for coach Willie Fritz to call a run up the middle and rely on his defense to protect a three-point lead.

The Cougars were out of timeouts, so at least another 45 seconds would have bled off the clock rather than just a few on an incomplete pass.

But Fritz went for the win, and quarterback Jonathan Banks and wide receiver Terren Encalade delivered the same way they had all game long.

Banks’ scrambling, 11-yard completion to Encalade near the sideline sealed the deal, and the Green Wave celebrated a huge 20-17 upset of the Cougars at Yulman Stadium.

“We didn’t want to get in one of those situations where we were trying to get a stop and hold on and all that other stuff,” Fritz said. “We’d had some success throwing, and it was good execution by the offense. Jonathan bought a little extra time getting out of the pocket, and it was a great catch.”

Tulane (5-6, 3-4 American Athletic Conference), which ended a four-game skid with an overtime win at East Carolina last Saturday, will play for its second bowl bid in the past 15 years next Saturday at SMU after winning on senior day for only the third time in the past 10 seasons.

Houston (6-4, 4-3), which beat top-10 Oklahoma and Louisville a year ago, suffered a tough loss under first-year coach Major Applewhite, a Baton Rouge Catholic graduate.

“It might be the best win (of my tenure),” Fritz said. “We needed a win like that. We’ve had some tough luck this season. I’m happy for the seniors and all the guys. We knew we were going to have to play (well) in all three phases. They are a talented squad.”

Banks and Encalade were terrific.

Banks completed 17 of 25 passes for a career-high 251 yards and also rushed 16 times for 68 yards. He scrambled for four critical first downs along the way, exhibiting the dual-threat ability that prompted Fritz to recruit him as junior-college transfer.

“One of our best plays is when he’s covered and takes off and runs,” Fritz said. “If we needed 10, he got 11. If we needed 8, he got 9.”

Encalade caught eight passes for a career-high 186 yards — the 12th-highest total in Tulane history — including the two most important plays for the offense.

His 64-yard catch-and-run 37 seconds after Houston rallied from a 13-3 deficit to go ahead 17-13 became the game-winner. He brought the ball in at the Cougars’ 46 on an inside route, made a quick cut to the outside and accelerated to the end zone as the Wave went ahead for good with 6:13 left.

His final catch, which he cradled near the sideline, was the clincher.

“I knew we needed a first down to come away with the victory,” Banks said. “They have a high-powered offense, and giving them the ball wouldn’t have been good.”

Banks took a knee on three straight plays to run out the clock.

“Every college football team and every NFL team likes the victory formation,” Encalade said. “That’s the best formation in football.”

Tulane was in control most of the way, scoring 13 straight points to close the first half after allowing a field goal on Houston’s opening drive. The Wave made big plays on both sides of the ball.

After safety Rod Teamer forced a fumble near midfield with a big hit on running back Duke Catalon, sending him to the locker room for the rest of the game, the Wave tied the score 3-3 on a 24-yard field goal.

After nickelback Jarrod Franklin and end Ade Aruna stuffed running back Mulbah Car on fourth-and-1 from the 6, Banks led a beautiful 13-play, 94-yard touchdown drive for a 10-3 lead in the second quarter.

In a theme that would be repeated the rest of the way, he hit Encalade for a 21-yard gain and scrambled for 9 yards to convert third downs along the way, leading to a 16-yard score from senior running back Sherman Badie.

Tulane finished 8 of 15 on third downs, controlling the ball for 36:24. Banks used a nifty move to escape a tackle and pick up 10 yards on third-and-9 the next time the Wave had the ball, setting up a 23-yard field goal as time ran out at the end of a 12-play, 77-yard drive.

“We had a good week of preparation, and we knew we had to win two more games to be bowl eligible,” Banks said. “We had to start off by beating Houston.”

A false start on tackle John Leglue at the Houston 1 that likely prevented a touchdown at the end of the half appeared as if it would prove costly when the Cougars went ahead. They got two touchdown runs from quarterback D’Eriq King, who burned Tulane for 141 yards on 10 carries.

The first touchdown came after a Banks interception gave the Cougars a short field. Houston earned the second one with a 91-yard drive that left Tulane in a familiar position — trailing at home in the fourth quarter against a good team.

At long last, the Wave knew what to do.

After Encalade’s huge touchdown, senior linebacker Luke Jackson knocking down King’s fourth-and-2 pass from the Tulane 32 with 2:41 left.

“We were lined up over the guard and had to have a quick drop and stop the quick slants they were pretty much running,” Jackson said. “(King) was kind of short (5-foot-11) and he couldn’t see me, so he just threw it directly at me.”

Follow Guerry Smith on Twitter, @guersmith