The world number four and second seed eventually imposed himself on his Argentine opponent after an even opening to the match to post a 7-6(2) 6-1 win in one hour and 26 minutes.
During the first set, Delbonis managed to save four set points before losing it on a tie-break but from then on in it was all about claycourt specialist Ferrer, who won six games in a row to seal victory.
Ferrer, who won the tournament in Buenos Aires at the weekend, moves on to meet Thomaz Bellucci in the third round of action in Brazil after the local wildcard downed seventh seeded Juan Monaco 4-6 6-3 6-3.
Fourth seed Robredo eased to a 6-2 6-1 success over Serbian qualifier Lajovic while eighth seed Andujar was handed a walkover when Klizan pulled out of their match, citing a stomach problem.
Rafael Nadal is the top seed at this ATP 500 event.
Haas had opened the tournament with a breezy win over Wayne Odesnik on Tuesday but the 35-year-old could not repeat the feat against an aggressive young opponent who had to win two qualifying matches to make the tournament's main draw.
Johnson banged out 13 aces and did not fold when Haas mounted a rally and claimed the second set.
In the decisive third, Johnson held firm and dominated the tie-breaker to gain a huge victory in his young career.
"I've had a few wins I'll never forget, and this is one of them," Johnson said. "I played a good week in Dallas (Challenger) two weeks ago and I played two good matches in qualifiers (here)."
The win marks the second time this season Johnson has defeated a top-20 opponent, after he took down Kevin Anderson, who was ranked 20th at the time, last month in Auckland.
This is not the first time Haas has bowed out of the Delray as a top seed as he was also ousted in the first round in 2010.
"I struggled at times with the lights and my focus, but I'm not going to take anything away from (Johnson)," Haas said. "I hate to lose, but I'm happy for him."
In other action, Australia's Lleyton Hewitt was also sent home early after he retired because of a right shoulder injury.
He was trailing fellow countryman Marinko Matosevic 7-6.
Fourth seed Anderson also advanced when opponent Ivo Karlovic was forced to retire early in the match, while sixth seed Feliciano Lopez overcame a one-set deficit to beat Adrian Mannarino 4-6 6-4 6-2.
Marin Cilic easily knocked out Benjamin Becker 6-1 6-3.
Rio de Janeiro Open results
2-David Ferrer (Spain) beat Federico Delbonis (Argentina) 7-6(2) 6-1
Thomaz Bellucci (Brazil) beat 7-Juan Monaco (Argentina) 4-6 6-3 6-3
Delray Beach International results
Marinko Matosevic (Australia) beat 8-Lleyton Hewitt (Australia) 7-6(2) 0-0 (Hewitt retired)
6-Feliciano Lopez (Spain) beat Adrian Mannarino (France) 4-6 6-4 6-2
7-Marin Cilic (Croatia) beat Benjamin Becker (Germany) 6-1 6-3