Freddie Sears was given his first start, having made his debut off the bench in the goalless draw at Halifax in midweek.
And the former West Ham youngster did well to win a loose ball and tee up the hard-working Charley Kendall to fire a low shot into the net on five minutes to give Daggers a great start and end their long wait for a goal.
Kwame Thomas headed over from Ollie Harfield’s deep cross for the hosts on 11 minutes, with Harry Phipps having a header deflected behind at the other end, before Nik Tavares saw his header from the resultant corner comfortably saved.
Shots rallied, with Lorent Tolaj seeing an effort blocked by Tavares on the quarter-hour mark, and they drew level midway through the half when Cian Harries netted from the penalty spot, after Elliot Justham was adjudged to have fouled Tyler Frost.
Daggers had a chance to regain their lead before the half-hour mark when Tavares flicked on a free-kick, but Sam Ling fired straight at Jordi van Stappershoef.
And the Dutch keeper produced a great save just moments later to deny Kendall his second, before Omar Mussa and Tavares were shown yellow cards in quick succession.
Harries volleyed over during first-half stoppage time, with van Stappershoef denying Tavares from Ling’s cross with a fine save at the other end.
But Daggers were reduced to 10 men just before half-time when Ling was dismissed for bringing down Joshua Stokes as he looked to break.
Josh Rees and Lewis Page replaced Kendall and Mussa during the interval, with Sears picking up a yellow card within four minutes of the restart.
Shots took the lead moments later as the ball fell kindly for Josh Stokes to fire into the top corner, leaving 10-man Daggers with a mountain to climb.
Captain Justham then needed treatment after a collision with Thomas, but he was able to continue with Daggers not having a replacement keeper on their bench.
And the visitors were back on terms midway through the half when Sears struck with a superb finish into the top corner after combining with Frank Vincent, who was booked moments later.
Sears then needed treatment, with Justham producing an outstanding save when play resumed to keep out Theo Widdrington’s deflected effort and ensure his side remained on level terms.
Hill made way for Sydney Ibie in the final Daggers change nine minutes from time, but Shots struck in the last minute of normal tmie when Thomas headed home.
A mammoth 12 minutes of injury time were signalled, with Tommy Willard going close to a fourth for the home side.
Sears then put Ibie through, but he could not take advantage, before van Stappershoef foiled Sears from range.
Dagenham & Redbridge: Justham, Hare, Tavares, Hessenthaler, Vincent, Hill (Ibie 81), Phipps, Kendall (Page 46), Ling, Sears, Mussa (Rees 46). Unused subs: Francis-Clarke, Appiah-Forson.