As the day wore on Southee couldn't find another major supporter at the bowling crease and Sri Lankan middle-order batsman Thilan Samaraweera went about starting a rescue mission for the home side.
Samaraweera found a reliable wingman in No 8 Suraj Randiv and the pair put on an unbroken stand of 97 for the seventh wicket as Sri Lanka eventually avoided the follow-on before the umpires called time due to fading light with around 20 overs left to be bowled in the day.
When stumps was called Sri Lanka was 225-6, with Samaraweera on 76 and Randiv on a test-best 34, and Sri Lanka will begin day four 187 runs in arrears on the first-innings but will have the belief that they can push towards knocking off more of that deficit.
In Samaraweera they have an experienced campaigner, who, at 36, is playing in his 78th test and also has made more than 5000 runs in the longest form of the game with 14 centuries.
New Zealand struggled to find ways to breach Sri Lanka's steadfast defence as the home side's batsmen were resilient and showed great patience at the crease as their total ticked over at a slow rate of 2.60 runs per over.
Southee finished the day with 4-51 from 19 overs and the 23-year-old appears to be showing the consistency that has been lacking from his game during the past year.
He took 4-46 in the first innings of the first test last week and is growing in to the spearhead of the pace attack that so many people predicted he would be when he first came on the international scene in 2008 as a 19-year-old.
Spinners Patel and Todd Astle were miserly but failed to genuinely threaten to take wickets and given how much turn the wicket was taking at the P Saravanamuttu Stadium that would have been disappointing for Kiwi skipper Ross Taylor.
There was also an issue with the balls being used and some estimates suggested that regular replacements of the shiny red things had seen no fewer than eight balls used in the test match during the first three days, presumably because they had lost their shape.
