The detection and accurate astrometry of fast-moving near-Earth objects (NEOs) has been a challenge for the follow-up community. Their fast apparent motion results in streaks in sidereal images, thus affecting the telescope's limiting magnitude and astrometric accuracy. A widely adopted technique to mitigate trailing losses is non-sidereal tracking, which transfers the streaking to background reference stars. However, no existing publicly available astrometry software is configured to detect such elongated stars. We present Astreaks, a streaking source detection algorithm, to obtain accurate astrometry of NEOs in non-sidereal data. We validate the astrometric accuracy of Astreaks on 371 non-sidereally tracked images for 115 NEOs with two instrument set-ups of the GROWTH-India Telescope. The observed NEOs had V-band magnitude in the range [15, 22] with proper motion up to 140$^{\prime\prime}$/min, thus resulting in stellar streaks as high as 6.5$^\prime$ (582 pixels) in our data. Our method obtained astrometric solutions for all images with 100% success rate. The standard deviation in Observed-minus-Computed (O-C) residuals is 0.52$^{\prime\prime}$, with O-C residuals <2$^{\prime\prime}$(<1$^{\prime\prime}$) for 98.4% (84.4%) of our measurements. These are appreciable, given the pixel scale of $∼$0.3$^{\prime\prime}$ and $∼$0.7$^{\prime\prime}$ of our two instrument set-ups. This demonstrates that our modular and fully-automated algorithm helps improve the telescope system's limiting magnitude without compromising astrometric accuracy by enabling non-sidereal tracking on the target. This will help the NEO follow-up community cope with the accelerated discovery rates and improved sensitivity of the next-generation NEO surveys. Astreaks has been made available to the community under an open-source license.