Leaders Birstwith got the better of Goldsborough, chasing down their hosts' first-innings score of 175/9 from 45 overs.
Mark Wood made 36 of those runs as Sam Barker took 3-26 for the high-flyers.
The in-form Craig Robinson then struck his third unbeaten half-century in as many appearances this term, leading Birstwith to 176/3 with plenty of overs to spare.
He finished on 79 not out after Sam Ryan had been dismissed for 50.
Newly-promoted Killinghall posted 159/9 at Helperby and that total proved beyond Dan Marston's men.
Luke Haidar led the way with a quick-fire 47 as Tom Messenger snapped up 4-39 and Simon Taylor bagged 3-24.
But, none of Helperby's batsmen could muster more than opener Luis Rees-Hughes' 28, and their reply closed with them 23 runs short.
Hamish McIntyre was the pick of the Killinghall attack, finishing the afternoon with figures of 4-38.
Routed by West Tanfield on the opening day, back-to-back wins have lifted Harrogate 3rds up to third place in the table.
On Saturday, Archie Holtham and James Culley snared three victims apiece as rock-bottom Blubberhouses were knocked over for 138.
Umaid Sajjad (59) was responsible for hitting a significant proportion of the home side's runs and he then impressed with the ball, claiming 3-40.
Lee Platts (4-33) was in even better form, however the Roosters eventually made it over the line with two wickets in hand, thanks largely to a fine half-century from Charlie Holtham (53).
Defending champions Darley suffered a second loss in three outings as they were edged out in a low-scoring affair at West Tanfield.
Larry Audain's 34 was the best any of the visiting batsmen could manage in their innings of 117 all out.
Jake Stubbs and Angus Shaw each sent a trio of Darley players back to the hutch, leaving Tanfield in a position of real strength at the halfway stage of the contest.
But, they made hard work of knocking off the required runs, struggling to 120/9 in the final over.
Audain completed a fine all-round display by helping himself to four of the wickets to fall.
Masham registered their maiden success of the campaign when they travelled to Pateley Bridge.
Joe Preece (88) performed superbly with the willow in hand alongside Chris Langley (44), steering the Badgers to 197/9.
Oliver Ambler (4-45) shone for Masham, who then required all but five balls of their allotted 45 overs to reach their victory target.
Ambler again did the damage, weighing with an unbeaten 63, while skipper Craig Broadley contributed 37.
Tom Hardcastle bowled well for Bridge to record figures of 3-27.
Kirk Deighton made light work of overhauling Arthington's score of 143/9.
Captain Louie Toseland opened the batting and smacked 62 not out during a first-wicket stand of 104 with Henry Blythe (40) as the visitors eased to 144/2 inside 38 overs.