Trapattoni left his post after last month’s defeat to Austria left them needing a mathematical miracle in order to secure passage to next year’s finals.
Current Under-21 manager Noel King has been placed in charge for their final two qualifiers, but his reign got off to the worst possible start when Sami Khedira opened the scoring for the hosts with a deflected effort on 12 minutes.
Ciaran Clark hit the bar just before the break as Ireland rallied, but Chelsea forward Andre Schurrle’s strike 12 minutes into the second half all but sealed the win.
Arsenal’s Mesut Ozil then continued his fine start to the season and put a gloss on the scoreline with the third in stoppage time.
The win means that Joachim Low’s Group C leaders cannot be caught with a game to spare after taking their points tally to 25 from a possible 27.
However, the game might have turned out very differently had Ireland, for whom goalkeeper David Forde was superb, enjoyed slightly better fortune in a flurry of activity either side of half-time.
Defender Clark, the man who was clipped by Khedira’s goal-bound shot, almost made amends in first-half injury time with a header which came back off the crossbar, and lone striker Anthony Stokes caught the Germans cold just 17 seconds into the second half with a shot which flew just wide.
The Republic at least restored a measure of pride a day short of a year since they were taken apart by Germany at the Aviva Stadium in a 6-1 rout which very nearly cost Trapattoni his job during the days which followed.
His successor King retained only four of the men who started against Austria last month, although he had no choice over suspended duo Richard Dunne and John O’Shea or the injured Robbie Keane, and only skipper for the night Seamus Coleman and James McCarthy survived from the mauling in Dublin.
He had worked all week on a system designed to frustrate the runaway group leaders and, while his 4-2-3-1 formation was not a surprise, the personnel he asked to fill some of those roles was.
Darron Gibson and Marc Wilson sat in front of the back four with a midfield trio of Kevin Doyle, McCarthy and Glenn Whelan playing behind lone striker Stokes.
The Celtic frontman was inevitably isolated for long periods, but Ireland’s early focus was on solid defence, and the plan worked.
Or at least it did for 12 minutes until their luck ran out.
Stokes was guilty of giving the ball away too easily deep inside his own half and German skipper Philipp Lahm pounced to feed Khedira, whose shot took a decisive deflection off Clark and completely wrong-footed Forde.
But it was not until the final 10 minutes of the first half that the pressure started to tell.
Schurrle, who had earlier seen appeals for a penalty waved away after he appeared to get a nudge in the back from Whelan, powered a header towards goal only to see Forde make a fine reaction save, and the Millwall man went one better eight minutes before the break to turn away Thomas Muller’s piledriver.
But the Republic were very nearly back in it in first half injury time when Gibson played a free-kick short to Whelan, whose cross was headed against the bar by Clark.
Ireland returned in determined mood and went close twice within the opening two minutes of the second half, Stokes firing wide after running on to Doyle’s knock-down and then forcing Manuel Neuer to tip over a dipping cross.
But the game was over when Schurrle controlled Kroos’ deft 58th-minute pass and beat Forde with ease.
Ireland might have got themselves back into the game with 17 minutes remaining but for a good double save by Neuer to deny first Stokes then Coleman.
However, Ozil had the last word when he ran on to Kroos’ pass in stoppage time to complete the victory.