Finnbogason struck after 45 seconds, which set the tone for a disappointing display from the hosts.
Altintop confirmed the three points would be heading back to the WWK Arena after the break, sealing Augsburg's third win in a row.
Finnbogason fires FCA ahead
It couldn't have started much better for the visitors, as they went ahead inside the opening minute. Max Kruse was dispossessed in midfield and Augsburg broke against the disorganised hosts. Halil Altintop was on hand to play the killer ball, and Alfred Finnbogason continued his fine form by slotting through the legs of Diego Benaglio.
Wolfsburg began to settle and quickly controlled the possession. Daniel Caligiuri almost provided the spark the hosts required when his cross-cum-shot almost looped in over Marwin Hitz, only for the Swiss stopper to reach back and claw it out of the top corner. There was further pressure from the Wolves, but they were met by a stubborn rear-guard action from the visitors.
Kruse was looking particularly lively, alongside Caligiuri, and the two combined to create the best chance of the match. The former was the beneficiary of a superb lofted pass from the Italian winger, though Kruse couldn't convert and saw his volleyed effort come back out off the underside of the bar.
There was one further moment of concern for the traveling band of supporters before the break, however. Paul Verhaegh was forced off with injury, and subsequently replaced by Jeong-Ho Hong. That aside, Markus Weinzierl's side were performing superbly with Wolfsburg having struggled to cause panic in the visiting defence.
Altintop adds crucial second
Wolfsburg were pressing hard in the opening exchanges in the second half, though it was an ongoing struggle to break down the opponents' back-line. Free-kick opportunities were increasing in regularity as time wore on, with the Wolves still unable to take advantage as frustrations grew in the crowd.
The groans soon turned to moans when Augsburg extended their lead before the hour-mark. An initial long-range from Finnbogason shot swerved to throw Benaglio off balance, but he did manage to get a hand on the ball. Altintop reacted quickest before anyone else and, with the Wolves off-guard, he rifled home his side's second.
It could have been even worse for the hosts had Benaglio not made a terrific save to deny Finnbogason once more. Wolfsburg were trying to find a way back into the game and had plenty of shooting opportunities, only for almost every attempt to find the gloves of Hitz.
A further few half chances fell the way of Dieter Hecking's side although the damage had been done, and their hopes of returning to the Champions League were ended as a result. They now face a real battle to even make a Europa League berth, which looks unlikely with a trip to Borussia Dortmund on the horizon. Augsburg will welcome 1. FC Köln on Friday with their Bundesliga status almost secured once more; the Southerners sit nine clear of Eintracht Frankfurt.