The Blues found themselves level at half-time, but conceded two sloppy goals in the second period that left them with a mountain to climb.
Nathan Stainfield’s goal with just over 15 minutes remaining gave Trinity hope, but they remain second-bottom in the National League North.
“It’s a disappointing result,” Sinnott told the club’s YouTube channel.
“I do believe that it’s a game we could’ve got some points out of, but if you make basic errors they’re going to cost you goals. That’s what happened today.
“For their second and third goals, they were avoidable. Especially coming after half-time, it really is the old equation of shooting yourself in the foot.”
Gainsborough have the opportunity to bounce back from a fifth defeat in six, when they head to fellow relegation threatened FC United of Manchester on Tuesday evening.
Sinnott continued: “You have to pick yourself up again and obviously with a team that’s low down, it’s a little bit harder to do.
“That’s the frustrating aspect of it – a couple of bad errors have cost us points. You can’t switch off in football.
“There are still points there to be won to get us out of this situation which I believe we are capable of doing.”
The trip to Broadhurst Park is the first of two in a row on the road for Trinity, who then head to second-placed Harrogate Town next Saturday to complete a testing week.