New Zealand trails by 48 runs in the first innings having dismissing Sri Lanka for 244 earlier.
Latham was unbeaten on 111 at stumps, batting with BJ Watling on 25. The pair added a 70-run partnership for the fifth wicket.
Latham faced 184 deliveries and hit 10 boundaries.
The Black Caps resumed at 103-3 after tea and Henry Nicolls (15) was caught by Dhananjaya de Silva off spinner Dilruwan Perera with the score on 126.
Perera took two wickets for 76 runs in 25 overs.
After New Zealand went to lunch on one without loss, Sri Lanka got its first breakthrough when Perera drew Jeet Raval forward with a flighted delivery that turned, took the outside edge and was safely caught by Dhanajaya de Silva at slip.
Captain Kane Williamson joined Latham and the pair added 33 runs together. Williamson, who just scored four runs in both innings of the first test, showed promise with 20 but edged fast bowler Lahiru Kumara to Kusal Mendis at slip.
Latham and Ross Taylor shared 50 runs for the third wicket before Taylor stepped out to hit left-arm spinner Lasith Embuldeniya only to be caught at slip by de Silva.
Earlier de Silva scored his fifth test century to rally Sri Lanka from trouble and reach a competitive 244.
De Silva walked in to bat when Sri Lanka was 93-4 and saw quick wickets fall at the other end to push the total to 131-6. He survived at nine when fast bowler Trent Boult dropped an easy high catch off his own bowling; but he made good his luck and held the Sri Lanka innings together by hitting 16 boundaries and a six from 148 deliveries.
He shared 41 runs with No. 8 batsman Perera (13) and 43 runs with Suranga Lakmal (10) at No.9.
Sri Lanka captain Dimuth Karunaratne made 65 while opening the batting.
Fast bowler Tim Southee continued his good run from the second day, taking two more wickets to finish with four for 63. Boult had three for 75.
Sri Lanka won the first test by six wickets in Galle and looks to sweep the two-test series.