The Eagles were just 40 seconds from their third Betfred Championship win in the opening four games on Saturday when they conceded a penalty after Bradley Knowles pushed an opposition player, which allowed London Broncos to draw level at 20-20.
Converted tries from James Glover and Izaac Farrell had given them a slender 12-8 half-time advantage over their full-time opponents and put them in a winning position at the interval for a second consecutive weekend.
Aston said: “At this moment in time we are giving too much away; a couple of tries last week from errors, a couple of tries this week from errors.
"If we smarten that up we are going to be a strong unit.”
Sheffield Eagles looked to have survived a second-half Broncos fightback and snatched the win when Aaron Brown crossed the line with just two minutes left and Izaac Farrell added the extras to take the scoreline in a close-fought match to 20-18.
Farrell scored a penalty goal earlier in the second period following a foul on Knowles, who was in the thick of the action all afternoon.
"The mood in the changing room was quite dejected because they believe they could have won,” said Aston. “We have got to manage it a bit better.
"The boys are gutted because we knew we did just about enough to win the game and we let ourselves down at the end, which is also great for us because it means come Friday we will be hungry again.”
‘Tubbs’ revealed the team had trained for just two hours in the build-up to the game, the equivalent of a morning’s work for London Broncos.
"We have got a good group, there’s no doubt about it,” he added. “The squad are working really hard for one and other.
“The work ethic and attitude is bob on and you can build things with that.”
Batley Bulldogs are up next for the Eagles at the Keepmoat Stadium on Friday evening (7.45pm kick-off).