Stefano Okaka opened the scoring after just four minutes in Udine, finishing off a Seko Fofana cross.
Rolando Mandragora sealed a valuable three points with a volley just after the hour mark as Udinese gain some breathing space by moving four points above the relegation zone with just their third win of 2019.
Badu has been sidelined since he underwent a successful surgery in July last year after picking an injury during Udinese pre-season training.
The 28-year-old midfielder was initially billed to be fit to start the season but his recovery delayed, leaving him out for the past ten months.
He last saw competitive action on Sunday, May 6, 2018, during Bursaspor’s 2-1 defeat against Fenerbahce in the Turkish SupaLig.
Udinese are sitting 15th on the standings with 28 points after 28 matches.