Kaepernick, who only took over the QB role from Alex Smith earlier this season, set an NFL rushing record for a quarterback by gaining 181 yards, including running touchdowns of 56 and 20 yards. He also passed for 17-of-31 for 263 yards.
Kaepernick also threw a pair of touchdown passes to Michael Crabtree - but began shakily in his playoff starting debut, throwing an interception on his second pass of the game that was returned 52 yards for a touchdown by Sam Shields for a 7-0 Green Bay lead.
But he recovered quickly, leading the 49ers on an 80-yard drive that he capped off with a 20-yard touchdown run to make it 7-7.
The teams traded touchdowns during an action-packed first half, with the 49ers moving ahead 21-14 after turning a pair of Green Bay turnovers into touchdowns and took a 24-21 lead on the stroke of half-time with a 36-yard field goal.
It was all San Francisco after the break as they ran roughshod over the Packers, rushing for three touchdowns as they put the game out of reach.
After Green Bay tied the game at 24-24, Kaepernick took charge. The heavily-tattooed quarterback faked a hand-off and took off to his right, using his long strides to race untouched into the end zone for a 56-yard touchdown that gave the 49ers a lead they never relinquished.
The 49ers will now face the winners of the Atlanta Falcons and Seattle Seahawks in the NFC title game on Sunday 20 January.
San Francisco had a total of 579 yards on offense, 323 on the ground. The previous playoff QB rushing record was 119 yards by Michael Vick.
Green Bay QB Aaron Rodgers completed 26-of-39 passes for 257 yards and two touchdowns. He ran for a more modest 28 yards.
A "Harbaugh Bowl" remains a possibility after 49ers coach Jim Harbaugh and Ravens coach brother John advanced to their respective championship games, just one win each from the Super Bowl.
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