The most obvious example in English football is probably Ipswich Town's 1961-62 title. Under Alf Ramsey, they were promoted from the Second Division only the season before and went on to win their first and only top-level title, beating favourites Spurs and Burnley (post-Munich, post-Wolves dominance, pre-Shankly days, innit) and by all accounts completely baffling most opposition tactically. Ramsey left the following year for the England national team job, and Ipswich finished bottom of the leaugue during the 63-64 season.
This probably wouldn't fall into the unexpected catagory, but in reading a bit about different leagues i found out about Dundee United's 82-83 title under Jim McLean. Before i was born and that, but it's their sole championship to date. The bigger story, though, is probably how competetive they were in Europe around that time, under Jim McLean. During his tenure, they made eight Scottish Cup finals, four Scottish League Cup finals, reached the semi-finals of the European Cup, and final of the UEFA Cup, admittedly losing both, but knocking out sides such as Barcelona, PSV, Monchengladbach, Monaco, Anderlecht and Werder Bremen. Their 82-83 title was won with a then record points and goals tally.
Probably the most interesting is Hellas Verona's 84-85 Serie A title. After accusations of match-fixing, the Italian FA chose to introduce the random assignment of match officials, much to the annoyance of the big clubs. Verona won the championship, but the next summer the league dropped the new system, allowing Juventus to bang out a few more titles.