Neither Carnegie nor Coventry impressed as the hosts ran out winners at Butts Park Arena.
Yorkshire made a strong start but fly-half Will Maisey slotted a penalty to give Coventry the lead.
Carnegie scored the first try as captain Pete Lucock crashed through between the posts and Elder converted.
Soon after, a kick through from Bulumaku bounced up for Max Trimble and the winger crossed before another penalty made it 13-7 to Coventry at the break.
Coventry scored first after the interval, Charlie Beech crashing over against his old side and they led 20-10 with 10 minutes to play.
The final score came from centre Andy Forsyth, who beat three defenders from 40m out.
Chris Stirling, Carnegie’s director of rugby, claimed: “We’re moving in the right direction, we’re not far off.
“The thing that hurt us the most was the fact that we were really limited in what we had on the bench. We needed to stay with the players on the field and at the back end of the game they were pretty much out on their feet.”
On new players Jade Te Rure and Antonio Kiri Kiri, he said: “I thought Antonio was really strong in the first 50, then lost his way, probably with jet-lag. Jade had some nice little touches but needs to sharpen up in accuracy.”
Coventry Rugby: Fenner; Bulumakau (Tolmie), Bulumakau (Stevens, 77), Stevens (Tuitupou, 57), Trimble; Maisey, Kessell; Gibbons (Alo, 57), Nilsen (Tolmie, 46), Jeannot (Beech, 43); Faleafa (Gray), Oram; Peters, Ram (Makaafi, 50), Nutley (Ram,55) (White, 74).
Yorkshire Carnegie: Elder; Watkins (Bullough, 19), Forsyth, Lucock, Brown; Te Rure, Bruzulier (Fox, 77); Thomas (Foster, 41), Buckle, Mitchell; Myerscough, Smith; Kiri Kiri, Bainbridge, Temm. Unused: Gibbings, Ilnicki, Frost, Brown.