Lacy finished with 82 yards, while tight end Jermichael Finley had a 10-yard touchdown catch in the first quarter before leaving with a neck injury.
Green Bay (4-2) won its third straight and took over first place in the NFC North.
Then Finley left with about 10 minutes to go with a neck injury after a late hit by Brown safety Tashaun Gipson on a 10-yard gain. He was carted off the field, and the team later said Finley had movement and feeling in his extremities.
The defense took care of the rest against a struggling Weeden, who finished 17 for 42 for 149 yards. The Browns' Jordan Cameron caught a 2-yard touchdown pass in the fourth quarter to cut the lead to 11 with 6:09 left.
Cleveland thought it caught a huge break after recovering an onside kick, only for the play to be erased by an offside penalty.
The Packers got the ball back, and Rodgers found Jarrett Boykin for a 20-yard touchdown and a 31-13. Boykin, making his first start at receiver with Jones and Cobb out, finished with eight catches for 103 yards.
The Browns did have two nice fourth-quarter kickoff returns that helped set up drives into Green Bay territory, including an 80-yarder by Travis Benjamin that led to Cameron's score.
Otherwise, bad luck lingered over the Browns like the dark clouds that drenched Lambeau Field with a steady shower most of the game.
On an earlier drive, a fourth-down pass from the 31 to Josh Gordon was broken up after Davon House reached up to knock the ball loose at the last second. House also had the first-quarter interception of Weeden.
Still, Green Bay penalties kept the Browns in the game well into the second half. Tramon Williams' sack of Weeden on a corner blitz was negated by 15-yard face mask penalty on A.J. Hawk. But Cleveland settled for a 44-yard field goal from Billy Cundiff to make it 17-6.
Penalties had bogged down the Packers after jumping out to a 14-point halftime lead. Good thing they got off to such a hot start.
Rodgers connected with Finley for a 7-0 lead after the athletic, 6-foot-5 tight end spun around one defender, then caromed off two others like a pinball into the end zone. A Lambeau leap was next, of course.
Lacy followed on the next drive with a 1-yard score to give Green Bay a two-touchdown lead.
It looked as if the Packers were pulling away for an easy victory, especially with Cleveland receivers dropping passes in the first half on a dreary evening.
The Browns couldn't muster anything on the ground, either, against the Packers' tough run defense. Willis McGahee had 11 carries for 39 yards