The back-to-back shutouts were the D-backs' first since June 3-4 2011.
"You always want to keep it going," Hellickson said. "You just never want to be that guy to go out and end something like that."
The D-backs scored more than three runs just once during the last seven games, but still won four games during that stretch thanks to performances like Hellickson's. The right-hander gave up just four hits and struck out six for his seventh win of the year and the first in his last three starts.
Hellickson has pitched six or more innings and allowed two or fewer runs in seven of his last 10 outings, including three of the last four.
"His off-speed stuff today was really good, changeup, breaking ball, it threw them off down at the bottom of the order especially," D-backs manager Chip Hale said. "He's able to throw pitches that aren't predictable."
The Brewers threatened early with runners in scoring position in four of the first five innings. But each time, Hellickson found a way to keep them off the board, including the first inning when he struck out Khris Davis and Scooter Gennett to strand a pair of runners.
"I made big pitches when I needed to," Hellickson said.
The Brewers scored in the ninth inning on Friday and since then, the D-backs' pitchers have combined to pitch 18 consecutive scoreless innings, their second-longest streak of the year. That included a strong eight-inning performance from Rubby De La Rosa in Saturday's 2-0 win.
In the last four days, D-backs starters have combined for a 0.33 ERA with 29 strikeouts and a .168 opponents batting average.
The D-backs started the 10-game homestand on a rough note when they were swept in three games by the Giants, then dropped two out of the three to the Marlins. They were able to salvage a 4-6 mark, though, carrying some momentum into a 10-game, three-city road trip that begins in Seattle on Monday.
"You can't expect guys to throw shutouts every night," Hale said. "Each guy just keeps stepping up. Now we go on the road to Seattle, a good team, we'll have to keep pitching that way and hopefully get a few more hits."