Kevin de Bruyne got a hat-trick of assists, linking up with Dost for both of the striker's goals.
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Bremen started the game brightly as Zlatko Junuzovic broke the deadlock in just under 10 minutes having found himself all alone in the Wolfsburg box to tap into an empty net after some neat wing-play from Levin Mete Oztunali.
However, Wolfsburg responded almost immediately as Daniel Caligiuri equalised just a minute later.
The goals then began to flood in as Franco Di Santo put the hosts back in the lead just four minutes later.
For the second time in the match, Bremen's lead didn't last long, though, as Maximilian Arnold levelled the scores up once again.
Bremen took the lead going into half time, but failed to keep hold of it after the restart. Dost netted his first following a neat assist by De Bruyne to again make the scores all square.
The in-form Dutch striker then got his second minutes later to give the visitors the lead for the first time in the match.
Caligiuri also claimed his second of the match to round off the scoring, but the decision to allow the goal appeared to be incorrect as Dost was offside in the build-up.
The win means that Wolfsburg stay in second but close the gap on leaders Bayern Munich to just eight points.