White top-scored with 83 and Bovey added 60, but with no other batsman reaching double figures the hosts ended their innings on 186, all out with two overs to go.
Bourne reached 189-6 in the 40th over to ensure victory.
Bradley McGilloway (2-14) and Nick Bennett (2-30) led the way with the ball while Steve Kirkham and Bovey both claimed wickets.
Alford will finish second-bottom of the county’s top flight, concluding their season at Louth, who sit a place and 36 points ahead of them, on Saturday (noon).
The Seconds host Brigg Town (1pm) while, the following day the East Lindsey XI host Louth Taverners (1pm).
Alford Ladies suffered double defeat against Hartsholme on Sunday in the Lincolnshire Women’s Cricket League.
Game one saw Hartsholme end their 20 overs on 116-3, wickets taken by Teresa Tate, Diane Hales and Suzanne Kelleher.
Joanne Duncan’s 21 led the way as Alford ended their innings on 69-4.
In game two, Alford were dismissed for 67, while Hartsholme responded with 68-1.
The wicket was taken by Shona Calthorpe.