In the first innings, Sri Lanka's seamers had had New Zealand at 188 for 6, before an outstanding Daryl Mitchell hundred, and lower-order hitting from Matt Henry pushed the hosts into a narrow lead.
"We've got the fast bowlers to exploit the conditions," Mathews said after play. "We've got some fantastic fast bowlers in the group, and we have the belief that if the batters get the runs on the board, the fast bowlers will definitely come into play, with the conditions. They've done exactly that.
"We've given ourselves a great chance to win the Test match. We have to turn up tomorrow and just go for it. The game is evenly poised. Latham and Williamson - we all know they are world class. We need to strike early to get into the game. If we can open one end, we can put a lot of pressure on the Kiwis."
The surface did not appear especially treacherous on day four - though there was still some movement off the seam for the quicks. The Hagley Oval surface has in the past tended to get lower and slower as a match goes on, though spinners have been effective at this venue late in the game as well.
"You can't predict a wicket 100%, but there was variable bounce today," Mathews said. "But hoping he [Prabath Jayasuriya] can hit the rough spots. Especially to the left-hander there's a big rough on either side. The spinner also might come into play - you never know."
His own 115 off 235 balls, which was his 14th Test ton and his second in successive tours of New Zealand, Mathews put down to experience. It was a vital innings, during which he forged a 105-run partnership with Dinesh Chandimal, then put on 60 alongside Dhananjaya de Silva.
Although in the first innings Sri Lanka rattled along at almost four an over, they were prevented from scoring freely for much of day four, by some disciplined New Zealand bowling.
Scoring was especially difficult before lunch, when New Zealand sent down 28 overs and conceded just 67 (a run rate of 2.39).
"You come set to play certain shots because they aren't giving anything away," Mathews said. "You kind of know what you get from each bowler, so you kind of mentally plan it out. The first session they gave nothing away at all. We had to keep fighting for sngles and twos - forget about the fours. We all know [Tim] Southee is a world-class bowler, and no matter how tired he is he lands it on the spot. Matt Henry bowled extremely well, as well as [Blair] Tickner though he's young and new to the Test arena, he bowled with a lot of gas. We knew what's coming with each and every bowler, and we had to plan accordingly."
"The captain and the coach play a major part in the team's environment. And us seniors will back it up with them. Chris Silverwood and his support staff and the captain has done a fantastic job in creating a great environment, to play cricket with a lot of freedom. That's what you want - to put everything aside, and go out there and enjoy yourselves, which we're absolutely doing."
Andrew Fidel Fernando is ESPNcricinfo's Sri Lanka correspondent. @afidelf