The first-team coach said they remain handily placed as they aim to win a hat-trick of Championships.
Yorkshire slipped to fifth in Division One after losing by an innings and four runs at North Marine Road.
But they are just 15 points behind Middlesex, who climbed to the top, and have a game in hand on the leading three sides.
“We were obviously disappointed to lose, but we’re less than 20 points from the top of the ladder,” said Gillespie.
“We’ve lost one out of eight Championship games this year, and we’ve won a couple of matches, so we’re still well-placed.
“I’m confident we can improve, and if we’re completely honest we haven’t been at our very best yet this season.
“But I can see a real spark in the lads’ eyes and a real desire, and I know they’re more than up for the challenge.”
After three days of sedate fare by the seaside, the game exploded into life on the final day. Middlesex reached 577 to gain a first-innings lead of 171 – the fourth-highest total in first-class cricket at the ground – and then dismissed Yorkshire for 167.
Yorkshire were put to the sword by the ninth-wicket pair of Toby Roland-Jones and Tim Murtagh, who added 107 runs inside 10 overs at the start of the day, before subsiding with the bat. Tim Bresnan top-scored with 39, one of only four batsmen to reach double figures.
“We played a decent game for the first three days, but there was a massive momentum shift on the final morning,” said Gillespie.
“You have to give credit to Middlesex – they came out and played incredibly positively – and we didn’t adapt quickly enough.
“With our batting, there were no demons in the pitch – we just didn’t score enough runs.
“But we dust ourselves off and move on to the next game.”
Yorkshire’s NatWest T20 Blast match against Derbyshire on Sunday has been moved from Chesterfield to Derby due to a waterlogged outfield at the Queen’s Park ground.