After rain delayed the start of play on the final day, at lunch the 274-run partnership between Mendis and Mathews had allowed Sri Lanka to erase all but nine runs of the deficit it labored under at the end of the first innings.
Mendis, who resumed at 116 not out, was 141 and Mathews, resuming at 117 not out, had moved to 120. Sri Lanka was 287-3, closing in on an overall lead with seven second-innings wickets remaining.
Rain delayed the start of play at the Basin Reserve by 90 minutes, allowing only an hour before lunch and returning just before the scheduled break to steal another over from the day.
Forecasts suggest the rain will become persistent and heavy during the afternoon, potentially aiding Sri Lanka to pull of one of the greatest escapes in test history.
Trailing by 296 runs after New Zealand made 578 in reply to its first innings of 282, Sri Lanka slipped to 20-3 in 12 overs before stumps on the third day.
Sri Lanka's position seemed hopeless and it seemed inevitable New Zealand would complete a win sometime on day four which would give it a 1-0 lead in the two-test series.
But Mendis and Mathews became the first batsmen to bat through an entire day in a test in New Zealand in a superb stand which will be responsible more than the rain for allowing Sri Lanka to save the match.
Tuesday was also the first full day in a test match anywhere in the world for almost a decade in which a wicket hasn't fallen.
Mendis' and Mathews' innings have been almost identical in quality and duration. By lunch, Mathews had batted 450 minutes — 7-1/2 hours — and Mendis eight minutes longer.
Mendis was slightly more willing to play his in the short passage of play possible before lunch, adding 25 to his overnight score while Mathews added three.
The New Zealand bowlers remained at a loss to devise means to separate the pair. Tim Southee had bowled 25 overs at a cost of 52 runs with eight maidens without looking threatening.