“We’ve given ourselves a great chance to win the Test match,” Mathews said after his masterful 115, made from nearly six hours in the middle in Christchurch.
His innings rescued Sri Lanka from a tight spot and led the tourists to a second innings total of 302.
That set New Zealand a target of 285, and the home side were 28 for one by stumps after 17 overs, with Conway out for five.
Tom Latham was not out 11 with Kane Williamson on seven, with New Zealand requiring 257 on the final day. The highest fourth-innings score at Hagley Oval is 256 for eight in a drawn Test.
“We had to work extremely hard to get those runs and we’ve got the fast bowlers to exploit the conditions,” Mathews said.
“We had the belief that if the batters get the runs on the board the fast bowlers will come into play. We need to strike early (Monday) to get into the game and if we can open one end we can put a lot of pressure on the Kiwis.”
A further 257 runs to win at Hagley Oval. Tom Latham (11*) and Kane Williamson (7*) see the team to stumps Day 5 🏏… https://t.co/GAGBRdXiBR
— BLACKCAPS (@BLACKCAPS) 1678599438000
Sri Lanka need another nine wickets on the final day but their chances could be dashed by rain that is forecast for the morning.
New Zealand have only pride at stake but Sri Lanka, playing their first Test in seven months, are chasing their ticket to the World Test Championship final against Australia.
They must win in Christchurch as well as the second Test in Wellington to make the final.
Sri Lanka also need Australia to win or draw their current fourth Test against India.
There was jubilation in the Sri Lankan camp when Kasun Rajitha had Conway caught and bowled for five, and similar scenes when he thought he had Williamson lbw without scoring.
A word from Blair Tickner after taking 4-100 today in his 2nd Test 🏏 #NZvSL https://t.co/oOtlz7pX99
— BLACKCAPS (@BLACKCAPS) 1678607508000
Sri Lanka reviewed the not out call but Williamson, who only made one in the first innings, was safe.
Mathews went to the middle late Saturday with Sri Lanka in trouble at 81 for three, which became 95 for four early Sunday with the dismissal of nightwatchman Prabath Jayasuriya.
But New Zealand’s confidence was dented as the 35-year-old Mathews put on 105 for the fifth wicket with Dinesh Chandimal and added a further 60 with Dhananjaya de Silva before he was dismissed by Matt Henry.
His 235-ball innings, filled with chanceless ones and twos and 11 boundaries, took Sri Lanka to a position of strength.
De Silva continued the momentum with his late cameo, finishing unbeaten on 47 while Rajitha and Lahiru Kumara chimed in with scores of 14 and eight as Henry mopped up the tail.
Blair Tickner, who accounted for the Sri Lanka top order, took four for 100 while Henry finished with three for 71.
New Zealand seamer Neil Wagner, who left the field injured on Saturday, did not bowl after scans revealed a bulging disc in his back and a torn right hamstring.
Wagner has been ruled out of the second Test starting on Friday.