Juventus midfielder Emre Can was sent off after just 14 minutes of the Group C match in Tallinn for a professional foul.
However, Joachim Low's injury-depleted squad regrouped and took the lead early in the second half through Gundogan's angled drive.
The Manchester City midfielder quickly added a second, with substitute Timo Werner knocking in a third as Germany moved level on points with leaders Holland and three clear of Northern Ireland.
Veteran goalkeeper Manuel Neuer replaced Marc-Andre Ter Stegen to captain Germany, while Gundogan and Marco Reus were the other changes from the 2-2 draw against Argentina.
In-form Bayern Munich winger Serge Gnabry had to be content with a place on the bench.
Germany made a bright start, with Can just unable to get on the end of a teasing cross from Gundogan inside the Estonia penalty area.
The Juventus midfielder was, though, soon dismissed.
After allowing the ball to run across him as he turned on the edge of the penalty area, Can made a rash sliding challenge to trip Frank Liivak as the Estonia forward looked to get clear on goal.
Bulgarian referee Georgi Kabakov immediately reached for the red card, much to the dismay of the former Liverpool man.
Estonia – thrashed 8-0 in Mainz just four months ago – almost took full advantage, as defender Joonas Tamm put the ball over at the back post following a corner.
Germany, though, soon regrouped and came close to breaking the deadlock themselves in the 24th minute.
Freiburg forward Luca Waldschmidt drove at the Estonia defence before cutting inside to drill a left-foot effort towards the bottom corner, which was pushed wide by goalkeeper Sergei Lepmets.
Estonia captain Konstantin Vassiljev failed to connect properly with an acrobatic attempt to meet a ball in from the right, with Neuer gathering the tame shot.
Reus was then unfortunate not to open the scoring five minutes before half-time when his 20-yard free-kick clattered off the crossbar.
However, Germany took the lead soon after the restart.
In the 51st minute, Gundogan latched onto a deflected clearance as the ball dropped across the edge of the penalty area to drill a fierce angled shot into the bottom left corner.
Before Estonia could regroup, they fell further behind just five minutes later.
Reus was the provider with a clever backheel inside the penalty area, which saw the ball run for Gundogan to plant another first-time effort past the keeper, albeit with a slight deflection.
In the 71st minute, Werner added a third not long after coming on.
The RB Leipzig frontman cut back inside to score from close range after being picked out by a sweeping cross-field pass by Gundogan.