as you said it yourself, the right coach and players, who after dominating in Italy for several years, had gained the winning mentality they lacked at Inter from previous years, being there at the same time gave them that CL-level mentality they needed to edge strong teams like Barca, Chelsea, etc. - adding players like Eto'o, Lucio, and even Sneijder, players with considerable CL experience, didn't hurt either. ... so, in the end, Inter did need to add that extra "CL factor" to win the tournament ... a cutting edge for the big games, of sort, that Bayern and Chelsea lack right now.
Of course, not getting
several PK being called against them and scoring a goal vs Barca from an offside helped too
There is one other important factor left, IMO - Barca's complacency after having won the CL and Spanish title the previous season ... which, IMO, also played a big role in Inter edging out Barca - Inter played those games like their life depended on it, while Barca didn't.