West hit Kupp for a 40-yard score in the second quarter and a 4 yarder in the third, the 54th receiving touchdown of his career, four behind the FCS record.
Miquiyah Zamora had a career-high 16 tackles for the Eagles (6-2, 5-0 Big Sky).
Josh Kealamakia's second field goal put Weber State (4-5, 3-3) up 13-7 just before halftime but he missed a 29-yard attempt in the third quarter and 38- and 40-yard attempts with the wind in the fourth. The last came with 2:16 to play.
Weber State had 27 first downs, 412 yards and a 23-minute advantage in time of possession.
For Eastern, which had 10 first downs and 189 yards, it was the lowest-scoring win since an 8-5 victory at Idaho in 2003.