The first chance fell Luton's way inside three minutes when Alfie Doughty's cross was headed over by striker Lasse Nordas, who was making his first start for the Hatters.
Luton's pressure continued and the game settled into something of a pattern, but despite dominating possession in the first half, no shots on target told a frustrating story.
Just past the 30-minute mark, against the run of play, Plymouth thought they had opened the scoring. Callum Wright met Tymoteusz Puchacz's pass and flicked the ball past Thomas Kaminski, but a tight offside decision went against the visitors.
Chances remained at a premium for both sides before the break. Nordas was heavily involved and went close with a cleverly improvised volley, but could not get it on target.
Another Luton chance went begging a minute after half-time when Elijah Adebayo blasted over the bar when the ball fell kindly in the area.
Adebayo was soon replaced by Brown, who opened the scoring and ended Luton's near seven-hour wait for a home goal with his very first touch.
He met Nordas' flick-on from Doughty's searching ball forward, before heading past Conor Hazard.
But Plymouth stayed in the game and, 20 minutes before full-time, Talovierov levelled. The Ukrainian defender headed home Kornel Szucs' cross while Luton defender Mark McGuinness was off the pitch after receiving treatment.
That goal instantly changed the mood and the home side's nerves were palpable. Although they huffed and puffed during a lengthy period of stoppage time, Plymouth - who had defended well throughout - held on for a point.
Image source, Rex Features
