The Dutch side led 2-0 after 17 minutes, before superb individual strikes either side of half-time from David Goodwillie and Morten Gamst Pedersen drew Rovers level.
But the hosts scored twice more in the second-half to secure the spoils.
In an end-to-end encounter, the Go Ahead Eagles opened the scoring on 13 minutes.
Rovers failed to clear a cross from the right and Xander Houtkoop supplied a left-foot finish from five yards out.
Houtkoop then turned provider as the Eagles doubled their lead four minutes later.
A well-worked move ended with Houtkoop delivering an excellent cross, which was finished first time by local lad Joey Suk at the far post.
On 27 minutes, goalkeeper Mark Bunn limped off and was replaced by Jake Kean, who pulled off a great save to deny Marnix Kolder.
As half-time approached, Goodwillie hit the post with a diving header from a Pedersen cross, but a minute before the break, the Scottish striker pulled one back.
After a great challenge in midfield, David Dunn fed Goodwillie, who looked up before firing a fantastic 25-yard shot into the top corner of the net.
Two minutes after the restart and the comeback was complete. Following a foul 25 yards out, Pedersen unleashed a trademark left-foot free-kick into the corner of the net.
Rovers' revival was short-lived however, as Go Ahead Eagles restored their lead on 55 minutes. A shot from Houtkoop hit the post, beyond Kean's despairing dive to his right, and as it bounced back out, it hit the onrushing Grant Hanley and went into the back of his own net.
The visitors had a couple of good chances to equalise again, through substitute Simon Vukcevic and Goodwillie, before the Eagles wrapped up the win on 76 minutes when Kolder tapped home after a well-worked break.
Rovers are back in action against Dutch Eredivisie side NEC Nijmegen on Sunday afternoon.
ROVERS: Bunn (Kean 27), Henley (Ribeiro 46), Orr, Hanley, Morris, Rochina (O'Sullivan 77), Dunn (Vukcevic 65), Pedersen (Petrovic 60), M. Olsson (Nunes 66), Edinho (Jorge 34), Goodwillie.
Subs not used: Robinson, Dann, Givet, Lowe, Formica, Gomes.