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BALTIMORE -- The lopsided score suggests the Baltimore Ravens solved some of their glaring offensive issues after a lackluster start to the preseason.

After failing to notch a touchdown in their exhibition opener last week, the Ravens will gladly build on the welcome improvement.

Rookie backup quarterback Tyrod Taylor directed two fourth-quarter touchdown drives, the first ending with Jalen Parmele's go-ahead 10-yard TD run with 7:16 remaining, and the Ravens rallied for a 31-13 victory over the winless Chiefs.

"You work hard and you want to see your hard work pay off," said running back Ray Rice, who ended Baltimore's touchdown drought in its sixth preseason quarter. "It's been one of them camps where you have to jam a whole offseason into training camp. To see the younger guys step up and do their thing, it's amazing. We're just looking forward to hitting the season in stride."

Taylor later added a 5-yard TD run of his own, a dive to the right pylon that withstood a video review. He finished 5 for 11 for 88 yards.

"He's done as well as we could have expected so far," Ravens coach John Harbaugh said of Taylor. "He's a very talented young man. I like his poise. He's very intelligent. He's a very fast learner. He's a pinpoint passer, a very accurate passer. He's a player. When he got out of trouble, he made some plays."

Kansas City's Tyler Palko completed eight of 13 passes for 95 yards but produced 10 points in his three drives -- a tying 4-yard touchdown pass to Terrance Copper late in the second quarter and a 12-play march culminating in a tiebreaking 30-yard field goal by Ryan Succop with 6:58 left in the third quarter.

"We talked going into this game about doing some of the fundamental things a lot better across the board -- offense, defense and special teams -- and I thought we were able to do that a little better in more areas, which is a positive sign to me," said Chiefs coach Todd Haley. "We obviously have a long way to go."

The Ravens, who lost 13-6 at Philadelphia on Aug. 11, played Flacco into the third quarter before Taylor took over. Flacco completed 12 of 24 passes for 124 yards.

Chiefs starting quarterback Matt Cassel was 6 for 14 for 73 yards. But Kansas City, blanked 25-0 by Tampa Bay last Thursday, clicked on offense once Palko entered midway through the second quarter.

"We're just trying to get better every day, and I think we took a step in the right direction offensively," Palko said. "It's baby steps. You need to get better every day."

The teams traded first-quarter field goals for a 3-3 tie.

Kansas City got a break when the Ravens' Lardarius Webb muffed a punt, with Jackie Battle recovering at the Baltimore 15-yard line. That set up Succop's 31-yard field goal.

Flacco converted two third-down passes on the ensuing drive, which ended with a successful 49-yarder by Billy Cundiff.

Rice finally notched the Ravens' first preseason touchdown with 3:10 remaining in the first half, busting through a gaping hole and cutting right for a 26-yard TD to cap a six-play, 81-yard drive.

"I always tell the line, just get me to the safety. If you get me to the safety, it's my job to get them to miss. The offensive line did a great job," Rice said.

Flacco's 43-yard pass to newly acquired Lee Evans on a sideline route set up Rice's scamper, which put Baltimore ahead 10-3. Evans, acquired in an Aug. 12 trade with Buffalo, quickly got on the same page with his new quarterback.

"It's kind of like changing pitchers, going from a fastball pitcher to a curveball pitcher, something like that," Evans said. "It's just the timing of it and seeing it in a live game situation."

Kansas City answered with its first touchdown of the exhibition season, tying the game with 12 seconds to go in the first half on Palko's 3-yard pass to Copper, who got a step on cornerback Cary Williams in the end zone.

After Baltimore was forced to punt on the opening series of the second half, Palko moved the Chiefs to the Ravens' 11-yard line on 11 plays before Succop's field goal made it 13-10.

After Parmele's run put the Ravens on top, rookie Taylor scrambled for a 5-yard score with 1:59 left.

Anthony Allen scored on a 1-yard run with five seconds left and, after the game, Harbaugh defended his decision to add points in the final minute of a game where the outcome was already decided.

"This is preseason," Harbaugh said. "If this was the regular season, we'd have been on a knee. The idea in that situation is to give the young guys who work hard and are trying to make a football team ... a chance to show what they can do. I just felt like that was the right thing to do for our players."

Game notes

The Ravens played without LB Ray Lewis, who was excused to attend to an unspecified family matter. ... Succop missed a 46-yard attempt early in the second quarter, caroming a drive off the right upright. ... Kansas City's first-round draft pick, WR Jonathan Baldwin, did not play. The NFL Network reported that Baldwin sustained an injury to his wrist or hand in a fight with a teammate earlier in the week. Baldwin's agent, Rick Smith, refused to comment on the situation when reached by telephone. "Jonathan Baldwin did not play tonight due to injury. That's the extent about what I'm going to talk about," Haley said, calling the incident "family business."