Several quarter-finals were squeezed into Sunday's (AEDT) schedule alongside evening semi-finals after rain ruin Saturday's (AEDT) play, forcing Alcaraz to play twice along with Diego Schwartzman, who also progressed to Monday's (AEDT) decider.
Alcaraz first defeated No.1 seed and 2021 Wimbledon finalist Berrettini in three sets, triumphing 6-2 2-6 6-2 in two hours and two minutes.
The 18-year-old No.7 seed, who reached last year's US Open quarter-finals, backed that up with a straight-sets win over Fognini, 6-2 7-5.
"It means a lot to me," the Spaniard said afterwards. "I surprised myself to be able to win two matches in a row in the same day. It's tough to win one match, two is even tougher."
No.3 seed Schwartzman got past fellow Argentine Francisco Cerundolo 7-6 (7-4) 6-3 in a two-hour-and-two-minute semi-final, having won through earlier in the day 7-6 (7-3) 4-6 6-4 against Pablo Andujar.
The top two seeds will meet in the Delray Beach Open final in Florida after wins by Cameron Norrie and Reilly Opelka on Sunday (AEDT).
No.1 seed Norrie got past American Tommy Paul, the No.4 seed, 6-3 6-3 in one hour and 16 minutes. Norrie had endured a slow start to the calendar year, but will now play for a third ATP Tour title.
No.2 seed Opelka had a tougher outing, needing three hours to overcome Australia's John Millman 7-6 (10-8) 6-7 (3-7) 7-6 (7-4) in a match which only saw two breaks of serve.
Opelka, who won last week's Dallas Open title, sent down 16 aces and won 82 per cent of his first serve.