A club statement said Dortmund were "full of sadness" after one fan passed away and another had to be resuscitated following two separate incidents.
Fellow fans watched the rest of Dortmund's 2-0 win in virtual silence after news of the death spread through the stands.
The club said the second supporter was in a stable condition in hospital.
The club statement read: "During the game there were two tragic incidents in the stadium in which two spectators had to be resuscitated.
"Unfortunately one person did not make it and has died. We are full of sadness and the families are in our thoughts.
"We have received the news from the hospital that the condition of the second person is apparently stable."
Thousands of fans sang You'll Never Walk Alone during the match and following the final whistle as a mark of respect.
Dortmund manager Thomas Tuchel: "My personal sympathies, just as those of the entire team and the backroom staff, go to the families and loved ones of the affected persons. It was a terrible situation.
"We all know that there are more important things in this life than football. It was very difficult to stand on the edge of the pitch or on the pitch, not knowing what was going on but still having to fulfil our obligation of putting in the best-possible performance."
Mainz boss Martin Schimdt: "Naturally our thoughts are with the relatives after such tragic events, and we wish to offer our deepest condolences.
"I found it inspiring how 80,000 fans fell quiet as a mark of respect in just a few minutes. Such solidarity is impressive, everyone in the stadium came together as a unit. That is, sadly, not always the case."
A fan has died after falling ill during Borussia Dortmund's Bundesliga game against Mainz on Sunday.