The visitors ended five years of hurt in some style too, making 186-5 on the final day, with captain James Middlebrook leading them home on 37 not out.
On Sunday, after losing the toss and bowling first, Beds saw their hosts rack up a massive opening stand of 183 before Martyn Cull (90) was bowled by the experienced Middlebrook.
Fellow opener Thomas Huggins went on to make 168 though, although Beds had more success in the middle order, as Middlebook retained some control, claiming 2-50 from his 16 overs.
Colin Griggs also had success, with 3-82 from 21 overs, while Andy Reynoldson took 1-128 as Suffolk finished on 341-6 from their 90 overs.
Beds had looked in trouble early in their innings, falling from 80-0 to 91-3 as Charlie Thurston (48), Luke Thomas (29) and Callum Berrill (0) all perished.
That brought George Thurstance to the crease though and he put on a century partnership with Griggs (24).
It was when he was joined by Middlebrook that the runs flowed though, with the pair putting on 197 as Thurstance made a sparkling 171, with 20 fours and three sixes.
Middlebrook also reached three figures, out for 101 at almost a run a ball, while Reynolds was unbeaten on 40 as Beds made 462-9.
Suffolk were then dismissed for 305 in their second innings, with Reynoldson claiming 3-75, setting up Beds’ chances of a long overdue win.
They are next in action on August 16, hosting Northumberland at Luton Town & Indians CC.