The 22-year-old quick bowler bowled unchanged for 22 overs at Northampton and persevered through the pain of a turned ankle he suffered early in the spell.
Betts spent more than an hour after his successes with the ankle receiving treatment but he insisted that he would be fit enough to bowl in the second innings.
His fine day was only slightly tarnished by a last-wicket stand of 47 which took the home county to 144.
Durham's reply was one of consolidation and when bad light stopped play with 21 overs remaining they had reduced the deficit to 26 for the loss of Jonathan Lewis and Martin Speight.
Gloucestershire's Mike Smith grabbed his fifth five-wicket haul of the season as Nottinghamshire were bowled out for 294 at Bristol but the home side struggled and were 84 for 4 when rain ended the second day early.
Sussex's dreadful run with the bat continued when they decided to bat at Hove against Surrey. They stumbled to 102 for 7 when rain before tea brought about an early finish.
Sussex, battling to avoid the wooden spoon in both the championship and Sunday League, lost Toby Peirce to the second ball of the innings and Neil Taylor and Rajesh Rao were also back in the pavilion with only five on the board.
Peter Moores provided most resistance for Sussex with an unbeaten 28 as Martin Bicknell claimed 3 for 41 and Chris Lewis 3 for 31.
Graeme Hick registered his first duck of the summer as Middlesex restricted Worcestershire to 90 for 4 at Kidderminster after they had been dismissed for 252.
Hick, after facing 19 balls, turned a bat-pad chance to short leg off Angus Fraser. It was his first run-less innings of the summer from 44 attempts in all cricket.