The Englishman got off to a solid start on Court 18, taking the opening set 6-3, but he encountered tougher opposition in the ensuing sets.
The second set was a close affair, going to a tie-break, which the German edged 8-6. Struff continued to match Evans's tempo and the third set ended up deadlocked before the Briton claimed the breaker 9-7.
Evans appeared to pull a muscle late on, but played through the pain and took the decisive set 7-5, putting Struff to the sword with a vicious backhand on the winning point.
The result sees Evans reach the second round of Wimbledon for the first time in his career, joining compatriot Marcus Willis in the draw.