Argentina international Bergessio struck in the 14th, 52nd and 71st minutes - the latter two after Siena had lost Felipe to a red card - as Catania earned their first win in six to consolidate their place in the top half of the table.
Siena, in contrast, are in deep trouble near the foot of the standings after slumping to a third successive defeat and they are now five points adrift of safety with only three games remaining.
To make matters worse for Giuseppe Iachini's men, those three games are against top four sides Fiorentina, Napoli and AC Milan.
On Sunday, Siena simply had no answer to Bergessio, who almost opened the scoring in the 10th minute with a fine turn and shot which visiting goalkeeper Gianluca Pegolo did superbly to save.
Pegolo was well beaten four minutes later though as Bergessio put Catania ahead with a shot into the bottom right-hand corner following a fine assist from Lucas Castro.
Siena tried to respond with two long-range efforts from Alessio Sestu that failed to trouble home keeper Alberto Frison before the visitors suffered a major setback three minutes before half-time when defender Felipe was dismissed.
The Brazilian was shown a red card after handling the ball to prevent Pablo Barrientos from breaking clear.
It was largely all one-way traffic in the second half as Catania made the most of their extra man.
The home side thought they had doubled the lead in the 47th minute when Castro found the back of the net but the goal was disallowed for a handball.
Siena then briefly threatened through Setsu, who produced a fine save from Frison with a far-post volley, but almost immediately they found themselves 2-0 down as Bergessio netted from close range.
Bergessio almost completed his hat-trick just after the hour mark after breaking clean through, but Pegolo came to his side's rescue.
The 28-year-old hitman did not have to wait much longer to secure himself the match ball though as he notched his third 19 minutes from time after gaining the benefit of a marginal offside decision.