Poland coach Paulo Sousa left out several regular starters including top scorer Robert Lewandowski, but his side got off to a flying start as Swierczok fired them ahead with his first international goal in the fourth minute.
The striker side-footed the ball past Russia goalkeeper Anton Shunin from eight metres after Przemyslaw Frankowski set him up from a deep pass by Mateusz Klich, delighting home fans in Wroclaw's Miejski stadium.
The Russians were pegged back in the opening exchanges but drew level in the 20th minute as right back Karavaev tucked the ball past goalkeeper Lukasz Fabianski at the far post after a superb Aleksandr Golovin cross.
Klich fired over the bar from long range six minutes later before Swierczok's cross-turned shot hit the crossbar with the unrelenting pace providing plenty of entertainment in the first half.
The game fizzled out after the break as both teams made an array of substitutions and neither was able to create any clear-cut chances.