The Bavarians' title party was postponed last week when they failed to beat Borussia Monchengladbach, before they were dealt a further blow in midweek when they crashed out of the Champions League at the hands of Atletico Madrid.
But a sublime performance from Lewandowski ensured Pep Guardiola and his men finally had reason to celebrate.
The former Borussia Dortmund man opened the scoring early on from the penalty spot and doubled his side's lead just after the half-hour mark with his 29th Bundesliga goal of 2015-16.
Moritz Hartmann pulled one back for Ingolstadt late in the first half, again from the spot, but an impressive performance from the home side failed to result in an equaliser.
Bayern needed just 15 minutes to open the scoring through Lewandowski. Marvin Matip brought down Franck Ribery inside the Ingolstadt area, leaving referee Florian Meyer little choice but to point to the spot, and the Poland international coolly slotted home for his 40th goal of the season in all competitions.
Douglas Costa should have made it two halfway through the first half after being sent clean through by Lewandowski, but the Brazilian hesitated too long in front of goalkeeper Ramazan Ozcan, allowing Benjamin Hubner to rush back and make a last-ditch tackle.
Ingolstadt caused Bayern plenty of problems on the break, though, and Hartmann should have levelled the scores after a good combination with Dario Lezcano, only to see his shot from 10 yards go inches wide.
The hosts were made to pay for Hartmann’s miss in the 32nd minute and it was Lewandowski who again found the net. Xabi Alonso picked the striker out with a sublime ball in behind the Ingolstadt defence and he beat Ozcan with a calm finish into the far corner.
Lewandowski could have had a first-half hat-trick after beating the offside trap and rounding Ozcan, but Hubner once more came to the rescue for the hosts with a goal-line block.
Ingolstadt never gave up and Hartmann deservedly pulled one back in the 42nd minute, converting a penalty after he was fouled by Alonso.
David Alaba and Douglas Costa fired wide from dangerous positions early in the second half as Bayern looked to finish the game, while Thomas Muller was denied by Ozcan after a brilliant throughball from Thiago Alcantara.
Ingolstadt, meanwhile, continue to charge forward in search of an equaliser whenever possible and Hartmann was unfortunate not to net his second goal of the afternoon when he fired just wide in the 70th minute.
But Bayern held on to their narrow lead over the remainder of the game to lift their first major trophy of 2015-16.
Key Opta stats:
- Arturo Vidal won his fifth national championship in a row (four with Juventus and one with Bayern).
- Kingsley Coman celebrated his fourth national championship in a row – at the age of 19! (twice with Paris Saint-Germain, once each with Juventus and Bayern).
- Philipp Lahm celebrated his 200th Bundesliga win with Bayern Munich.
- Robert Lewandowski scored his 29th Bundesliga goal in 2015/16 and equalled the record of Klaas-Jan Huntelaar (2011/12) for a foreign goalscorer in the Bundesliga.
- Ingolstadt had 16 shots, a Bundesliga record in 2015/16 versus Bayern Munich.