The victories spiced matters up with the top two set to clash on Wednesday in Dortmund.
Bayern, who served up their ninth straight win, are the record title winners with 22 crowns but reigning champions Dortmund are in pole position for an eighth title with 66 points from 29 games with just five rounds of action remaining.
Dortmund took the lead in the 22nd minute with a right-footed effort from Robert Lewandowski set up by Ivan Perisic, and, four minutes after the restart, Ilkay Gundogan made it 2-0.
But Wolfsburg, in midtable and looking a pale shadow of the side who shocked the nation by lifting the championship in 2009, gave themselves hope with a goal on the hour mark from a Mario Mandzukic header.
However, the hosts were dealt a blow when Alexander Madlung was sent off late for a second booking and there was still time for Lewandoswki to bag his second goal of the game and make the points safe.
Dortmund coach Jürgen Klopp was a happy man afterwards.
“If you win 3-1 in Wolfsburg you should be satisfied and we are,” he said.
The win preserves the cushion Dortmund, held to a 4-4 draw last week by Stuttgart, needed ahead of Wednesday’s summit meeting with Bayern, who got off to a flier with Gomez on target after just 24 seconds, firing in at the second attempt after Franck Ribery’s cross had found him lurking.
Relegation-threatened Augsburg recovered from that early shock, however, and Koo Ja-cheol deservedly levelled in the 23rd minute after Axel Bellinghausen beat Philipp Lahm for speed down the left flank and crossed into the box for the South Korean to finish.
Gomez ended dogged Augsburg resistance on the hour when Lahm made up for his earlier error by feeding Arjen Robben.
The Dutch flier played in Gomez for a typically predatory finish and his 25th goal of the season to leave the battling visitors just a point above the drop zone.
Bayern’s veteran coach Jupp Heynckes said he knew that the game would be a tough test.
“The players needed a while to find their rhythm but then we got it together after the break and we can now travel to Dortmund feeling optimistic and motivated, even if that game itself won’t decide the title race.”
Klopp also had an eye on Wednesday and told reporters: “We’re really looking forward to the Bayern game – we want to offer a real spectacle for 80,000 fans, but (the game) won’t decide anything.”
Elsewhere, Stuttgart were among the goals again as they hammered Mainz 4-1 to stay fifth – albeit eight points behind Borussia Mönchengladbach, who were later hosting Hertha Berlin seeking a win to go level with third-placed Schalke.
Schalke – nine points behind Bayern – host Hanover on Sunday.
Cologne remain in the drop zone following a 1-1 home draw with Werder Bremen, who remain two points off Stuttgart in sixth as both chase European berths next season.
Kaiserslautern’s 2-1 loss to Hoffenheim means they are all but down as the tailenders, champions in 1991 and 1998, remain ten points from safety.