They was just a minute remaining of the eight signalled for stoppage time when Bradford were awarded a penalty after Terrell Forbes tripped winger Zavon Hines.
Substitute Alan Connell duly equalised from the spot, cancelling out Kieron Cadogan's first-half opener, to secure a 1-1 draw.
Nevertheless, Aldershot continue to gradually pull away from the relegation zone after extending their unbeaten run to six games, four of which have come under recently-appointed Scott.
He said: "When you come so close to winning and then concede a late equaliser is a disappointment, but we have got to take positives out of it. Bradford is a tough place to come and we have got to look at this as a point gained in our quest to stay up.
"I couldn't fault any of my players - everyone has given their all.
"The crowd got behind Bradford and carried them through, I thought they were fantastic.
"Bradford are a good side, look what they have done this season, and on their bench were four players who played in the League Cup final. So they have strength in depth, but we defended very well and we also knew they were vulnerable and to go in front just before time was a perfect tonic."
Scott, who lost two senior players to injury this week, also saw goalkeeper Jamie Young carried off after a collision in the 70th minute, which caused a lengthy delay as he was then taken to hospital, while skipper Ben Herd went off with cramp.
"Jamie was clearly struggling," Scott said. "He got knocked out in the collision and his legs were wobbly."