Alex Smith of the San Francisco 49ers drops back to...

Alex Smith of the San Francisco 49ers drops back to pass against the Cleveland Browns in the second quarter. (Oct. 30, 2011) Credit: Getty Images

Frank Gore charged up the middle with his signature burst for a 2-yard gain, taking another step forward in franchise history.

Gore got that coveted new contract and is playing like his old dominant self. Forget the hip injury that ended his season last November. He’s a new man making his case for a third Pro Bowl and pushing his team toward its first playoff berth in nine years.

Gore ran for both 125-plus yards and a touchdown in a fourth straight game, Michael Crabtree made his first TD reception of the season, and the San Francisco 49ers beat the Cleveland Browns 20-10 on Sunday.

Alex Smith completed 15 of 24 passes for 177 yards, leading the first-place 49ers (6-1) to their fifth consecutive victory since an overtime loss to the Cowboys in Week 2.

For Colt McCoy and the Browns, it’s 0-for the Bay Area. They lost two weeks ago in Oakland, then flopped across San Francisco Bay at Candlestick Park.

No handshake flap this time for Jim Harbaugh, either. And no post-bye letdown for his 49ers — even if it was far from flashy.

This marked Harbaugh’s first game since his seconds-long greeting and firm backslap of Lions coach Jim Schwartz following San Francisco’s 25-19 comeback win at Detroit two weeks ago.

Harbaugh says he practiced during the bye week by high-fiving with his toddler daughter. He was polite to fellow first-year coach Pat Shurmur when they met at midfield.

The 49ers — who were made aware by Harbaugh this week that teams were 3-9 after their byes coming into Sunday — are off to their first 6-1 start since 1998 in the days of greats Jerry Rice, Terrell Owens and Steve Young.

Cleveland’s D’Qwell Jackson committed a costly facemask penalty against Gore that would have stopped San Francisco on third down 2 yards short in the waning minutes. David Akers sealed it with his second field goal of the day, a 26-yarder.

Gore already had 103 yards on 19 carries by halftime, then got the other 7 yards he needed to pass Roger Craig and move into second place on the franchise rushing list with 7,089. Gore wound up with 134 yards on a season-high 31 carries.

San Francisco has a five-game winning streak for the first time since 2001 and wrapped up its first unbeaten October since 1997.

The next big task will be ending an eight-year stretch of frustration in which the team hasn’t had a winning season or made the playoffs. The Niners began the day with a comfortable three-game cushion in the division — nicknamed the NFC Worst last season, when Seattle won with a 7-9 record.

Harbaugh’s big brother, John, helped the sibling cause by rallying to beat Arizona 30-27 earlier Sunday.

The banged-up Browns (3-4) improved on special teams but went without their top two running backs. Peyton Hillis didn’t play for the second straight game because of a hamstring injury and Cleveland then lost Montario Hardesty to a calf injury in the first half.

McCoy and Co. managed only Phil Dawson’s 52-yard field goal on the final play of the first half before Josh Cribbs’ 45-yard TD catch with 6:17 to play.

After last week’s lackluster 6-3 win over the Seahawks, it didn’t get much better for the Browns. They failed to reach the end zone for seven quarters before Cribbs scored.

McCoy accounted for four of Cleveland’s five fumbles, one of which was lost and helped set up Gore’s early TD run.

McCoy lost the ball on the second play from scrimmage after he was stripped by Ahmad Brooks. Isaac Sopoaga recovered at the Cleveland 20. Gore ran 4 yards for the first points. Gore rushed for a touchdown in his fourth straight game, the first time in his career he’s done so.

McCoy was sacked four more times after five last week by the Seahawks.

San Francisco’s lopsided first half told the tale far before the clock finally wound down on a crystal-clear, unseasonably warm fall day.

First downs: 15-6. Total yards: 253-93. Time of possession: 18:58-11:02. Rushing yards: 145-25.

Crabtree caught a 2-yard pass to put the Niners ahead 17-0 just before halftime. Dashon Goldson made his first interception of 2011 in the end zone in the third quarter.

Left tackle Joe Staley even was in on the action, catching a 17-yard pass from Smith for a first down midway through the first quarter for his first career reception. He jumped up in celebration and signaled the first down by emphatically pointing down field with both hands. Sopoaga had an 18-yard reception late against the Browns’ fourth-ranked defense.

49ers wideout Braylon Edwards returned against his former club in his first action since right knee surgery after getting hurt against the Cowboys. He had four catches.

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