We investigate the problem of speaker independent acoustic-to-articulatory inversion (AAI) in noisy conditions within the deep neural network (DNN) framework. In contrast with recent results in the literature, we argue that a DNN vector-to-vector regression front-end for speech enhancement (DNN-SE) can play a key role in AAI when used to enhance spectral features prior to AAI back-end processing. We experimented with single- and multi-task training strategies for the DNN-SE block finding the latter to be beneficial to AAI. Furthermore, we show that coupling DNN-SE producing enhanced speech features with an AAI trained on clean speech outperforms a multi-condition AAI (AAI-MC) when tested on noisy speech. We observe a 15% relative improvement in the Pearson's correlation coefficient (PCC) between our system and AAI-MC at 0 dB signal-to-noise ratio on the Haskins corpus. Our approach also compares favourably against using a conventional DSP approach to speech enhancement (MMSE with IMCRA) in the front-end. Finally, we demonstrate the utility of articulatory inversion in a downstream speech application. We report significant WER improvements on an automatic speech recognition task in mismatched conditions based on the Wall Street Journal corpus (WSJ) when leveraging articulatory information estimated by AAI-MC system over spectral-alone speech features.