The 29-year-old scored four minutes into Bayern's third-round tie but had to be stretchered off 12 minutes later having been sent tumbling by Augsburg goalkeeper Marwin Hitz.
Robben received lengthy treatment from Bayern medical staff pitchside and the European champions said he had suffered a "deep cut" on his right knee.
The extent of the injury will only be known after a scan.
Robben has been in outstanding form for Bayern recently, scoring in each of his last four matches for the Bundesliga leaders.
His replacement Thomas Muller scored Bayern's second, heading home 12 minutes from time to put Pep Guardiola's side into the last eight.
Bayer Leverkusen, second in the Bundesliga, enjoyed a 2-1 win at Freiburg with Australia winger Robbie Kruse scoring for the visitors with just a minute gone.
Eintracht Frankfurt's ex-Real Madrid and Spain U-21 forward Joselu scored a hat-trick in their 4-2 win at home to second-division Sandhausen.
Wolfsburg needed Croatia striker Ivica Olic to hit the 89th-minute winner to seal their 2-1 victory at second-division Ingolstadt 04.
The hosts took a shock early lead before Wolves clawed their way back with Brazilian defender Naldo equalising with 20 minutes left.
On Tuesday, Borussia Dortmund advanced with a 2-0 win at minnows Saarbruecken while Ruhr Valley neighbours Schalke crashed out with a 3-1 defeat at home to top-tier rivals Hoffenheim.
The Royal Blues' defence was left in disarray as Hoffenheim raced into a 3-0 lead after just 35 minutes with goals by Kai Herdling, Kevin Volland and Roberto Firmino.
Schalke pulled one back through Peru forward Jefferson Farfan with just over half an hour to go.
Hamburg went through as they enjoyed a 2-1 win over second-division Cologne, while Union Berlin went down 3-0 at home to Kaiserslautern in an all second-division tie.
The quarter-finals will be played on 11 and 12 February with the final at Berlin's Olympic Stadium on 17 May.