Alastair Cook (77) and Jonathan Trott (15) helped the tourists end the day in control on 154-1, in reply to Sri Lanka’s total of 275.
Andy Flower’s side must win in Colombo to keep their status as the number one side in the Test rankings, and the Nottinghamshire spinner believes they are in a good position.
“To be on that score at the end of the day was just what the doctor ordered and it’s the perfect platform for us to build a big first innings lead,” Swann told the BBC.
“We need to use the whole of day three and maybe a bit of day four to build a lead that we can then defend.”
He added: “It’s going to be hard work to shift their exceptional middle-order so we need runs on the board. If I got my way, we would get 500 or 550.”
Sri Lanka resumed day two on 238-6, with England seeking to bring the innings to a rapid conclusion.
After failing to make any early inroads, Swann’s introduction certainly had the desired effect for Flower’s side.
First, Suraj Randiv was drawn into a rash shot to fall for 12, with Kevin Pietersen taking a good catch.
The Nottinghamshire man didn’t take long to strike again, as he removed Angelo Mathews for 57, just four balls after the day’s first dismissal.
Despite some defiant batting from the tail-end, the final two wickets were quickly claimed.
Rangana Herath (2) was caught behind after taking a wild swipe at a Tim Bresnan delivery, before Swann claimed his fourth victim shortly after, with Suranga Lakmal falling for a duck.
England’s openers made a strong start to the innings, as they looked to turn around a poor run of form.
Strauss’ failure to perform with the bat had put his place in the side under scrutiny, but the Middlesex man responded well to the pressure.
Alongside opening partner Cook, they built up a painstaking stand of 122, as the hosts were frustrated in their attempts to find early wickets.
After Strauss was dismissed for 61 by Tillakaratne Dilshan, Trott came in to ensure they ended the day 121 runs behind, ahead of a crucial third day.
MORE: The latest football news
MORE: The latest tennis news
BIOGRAPHY: Anthony Martial