Massive thunderstorms halted play three times during the week and after Saturday's round was washed away, officials decided to reduce the tournament to 54 holes.
Raphael Jacquelin and Nic Henning shared fifth at 12-under while Anthony Wall finished alone in seventh at minus 11.
Fichardt had a three-shot lead after the second round was completed on Saturday morning. He opened with a bogey at No. 2 after he hit a poor 9-iron shot and then failed to get up and down.
That would be the lone mistake in Sunday's final round for Fichardt. He birdied the fifth hole when he wedged his approach inside of 10 feet.
Rumford had started to close the gap as the players made the turn but Fichardt distanced himself starting at No. 7. He hit an 8-iron at the 188-yard hole and then watched as it landed in tap-in range, almost dropping in the cup. Fichardt reached the green in two at the par-5 8th and two-putted for another birdie and at nine he played a 7-iron into the back-left portion of the green and then drained a 15-foot downhill birdie putt.
Fichardt parred the next six and when Rumford double-bogeyed the 8th, the championship seemed to be over. Fichardt birdied 16 after he ran home a 25-foot bomb and then parred out for his first win on the European Tour.
'This is definitely a breakthrough for me,' said Fichardt. 'In the South African Open last year, I thought I had it in the bag and lost by one shot. But this is a breakthrough for me. I'm really happy about this.'
Fichardt won the inaugural Tour Championship on the South African Tour last month but this victory seems to be sweeter.
'This is a bit bigger than the win in South Africa last month,' said Fichardt. 'The prize money is four times bigger. That was a big win for me with (Ernie) Els and (Mark) McNulty in the field. It was a stepping stone for me at the beginning of the year.'
Rumford double-bogeyed the 8th but rebounded with four birdies in his final six holes. He posted a 2-under 69 on Sunday for his share of second.
Johnson fired a 6-under 65 thanks to eight birdies and two bogeys.
Coceres made a surge at the turn with two birdies and an eagle over three holes but a bogey at No. 14 coupled with Fichardt's birdie run in the middle of the round closed the door on Coceres' chances.
Click here for full-field scores from the Sao Paulo Brazilian Open