The only goal of the game came in the first half as Sullivan struck a superb effort to send Fishermen into the lead.
Neither side could add to their tally after the break in what proved to be a game of few chances and the away side held on to claim derby bragging rights.
Jordan Purvis was man of the match for the victors, while Charlie Smith stood out for Sleights.
In the only other game in Division One, a Luke Delve hat-trick fired Westover Wasps to a 3-1 win at Seamer.
Delve headed in twice to send his side 2-0 up before adding a third from close-range to make it 3-0 at the break.
Seamer huffed and puffed in the second half to find a way back into the game, but they could only manage a consolation strike from Danny Glendinning after 80 minutes.
Westover boss Jamie Heritage singled out Nathan Barber as his side's man of the match, while youngster Cameron Greatorex stood out for Joe Wright's Seamer side.
Filey Town Reserves edged closer to the Division Two title after hammering Falsgrave 8-0.
Aaron Howard, Tom Haxby and Billy Coyne all hit two apiece, while top-scorer Joe Gage and Max Gage added one each.
Howard and Joshua Westmoreland were Filey's star men.
Filey's most likely challengers, Seamer Reserves beat Scalby 3-1 at Carr Lane, despite the hosts leading at the break.
Player-boss Tommy Adams smashed in a double and Danny Kelly grabbed their other goal, while man of the match Craig Rackham struck for Sam Medd's Scalby.
Seamer stay in fourth, but have three games in hand on leaders Filey.
Edgehill Reserves stay in second after a 2-0 win at champions Newlands Reserves.
Club stalwart Andy Noon smashed in a stunning long-range strike and Jon Cairns also netted for Edgehill, who are seven points adrift of the league leaders.
FC Rosette picked up their first win of the season with a 4-1 victory at home to Duchess.
Dane Robinson hit two for Rosette to send them into a 2-0 lead before Al Wray added a third for the hosts.
JP Watson struck to ensure Duchess had hope at the break at 3-1, but they couldn't find a way back into the game in the second half and Danny Birley added a fourth to hand Rosette a 4-1 win.
Robinson was Rosette's man of the match.
Goalsports and West Pier Reserves played out a 1-1 draw.
Mark Plumpton's side led 1-0 at the break via Tyson Stubbings' goal, but the goalscorer had other chances to add to their tally and he spurned them.
The hosts' lack of clinical finishing came back to haunt them in the 86th minute when Christian Mattison's corner flew straight into the back of their net to seal a point for Jonny McGough's men.
Midfielder Sam Pallant was Goalsports' man of the match, while Jack Brown and Mikey Anderson shone for McGough's below-par Pier.
Edgehill 3rds kept up the pressure on Sherburn at the top of the Division Three table with an 8-5 win against a depleted Scalby Reserves side.
The hosts led 4-0 at the break after goals from Tom Scales, Kyle O'Toole, a Simon Robson penalty and Carl Hepples.
Despite going 5-0 down to a Dave Hooper pen, Scalby rallied back to 5-3 after Josh Williams hit two and Matthew Bourne struck.
Scales and O'Toole added a second apiece for Martin Cappleman's side, to make sure of the win at 7-3.
Scalby battled back again and added two more, Bourne hitting his second and sub Tom Crawford firing in, but Edgehill had the last word as Glen Gibson headed in an eighth.
Former Scarborough Athletic midfielder Gary Hepples ran the game for the victors, while young full-back Ben Dolan was Scalby's man of the match.
Eastway's Zachary Hansen smashed in five goals and added three assists as his side romped to a 10-3 win against Hunmanby United Reserves.
Dom Barber and Dave Darling both added two goals, while super-sub Barton Newham also grabbed a goal.
Goldsborough United consolidated their place in the top four with a comfortable win against Seamer 3rds.
Simon Taylor hit two for the victors, while man of the match Ben Watson and Ian Rhucroft also found the back of the net.
Ayton continued their resurgence with a 5-1 win against Fishburn Park Reserves.
Chris Milburn hit two for Ayton, and they cruised to the victory thanks to further goals from Chris Reddish, Tyler Beck and sub Dan Bywater.
Ayton's win moved them closer to the top six and they could leapfrog their visitors if they win their two games in hand.