A penalty from Adam Rooney and another strike from Niall McGinn put the Dons 2-0 ahead on aggregate before Maikel Kieftenbeld pulled a goal back a minute before the interval.
Aberdeen scored a crucial away goal when Rooney (26) calmly sent Sergio Padt the wrong way from the penalty spot after Kieftenbeld fouled Peter Pawlett.
The Groningen defence were opened up again on 33 minutes when Willo Flood sent Rooney clear, this time the striker lobbed the advancing keeper and when the ball came back off the crossbar McGinn slammed in the rebound.
Kieftenbeld gave the Dutch side some hope when he got between Russell Anderson and Shay Logan to meet a deep cross from Nick van der Velden, and despite Jamie Langfield’s best efforts the ball spun over the line.
After the break Anderson was caught underneath a cross and Van der Velden laid the ball off to Michael de Leeuw, but Langfield was quick to react and stopped the shot with his trailing leg.
And Aberdeen survived a penalty claim when Anderson appeared to barge into the back of De Leeuw.
But Ryan Jack came closest in the closing stages when his shot was deflected wide after a wonderful run from Jonny Hayes.
In a frantic start to the tie, Hayes was well placed on the line to clear the danger as a deep corner from Groningen was headed towards goal by Tjaronn Chery.
The visitors responded as Rooney immediately broke clear but was unable to beat Padt, and a minute later Filip Kostic shot wide of the Aberdeen post.
Padt was called into action again on 16 minutes when he smothered a Hayes shot at the second attempt with Pawlett ready to pounce on a rebound.
Chery came close with a first-time effort from 25 yards shortly after the second Aberdeen goal and a few minutes later Rooney almost got on the end of McGinn's flick.