Jóhann earned the International Master title in 1984 and the Grandmaster title a year later. Among his best international tournament results are shared first place at Reykjavík in 1984 and 1992; shared first at the World Open 1991 in Philadelphia; equal third at Tilburg 1988 (+3−3=8); and sixth at Belgrade 1989 (+2−2=7). In 1987 he finished equal first at the Interzonal tournament in Szirák, Hungary and qualified for the World Championship Candidates Tournament in 1988. He defeated Viktor Korchnoi but lost to Anatoli Karpov in the quarterfinals. Jóhann competed in the FIDE world championship held with the knock-out format in 1998 and 2004.