Wanderers, who were named among the favourites for promotion at the start of the season, looked devoid of ideas in attack throughout the match as they suffered their fourth successive league defeat.
The Trotters are still searching for their first three points of the campaign and remain level with Millwall at the bottom of the table. Leeds, meanwhile, are up to sixth.
Leeds' Michael Tonge and Bolton's Chris Eagles came in for their first starts since the opening day, in place of Paul Green and Jermaine Beckford respectively, with both players dropping to the bench.
Leeds opened the scoring for the first time this season in the seventh minute.
After Zat Knight cleared for a corner, Luke Murphy's delivery saw Varney get across his man and head the ball into the back of the net at the near post.
Bolton almost hit back after two efforts from David Ngog. The first saw the Frenchman get across Jason Pearce and steer the ball past Paddy Kenny, but just wide of the far post.
Darren Pratley then found Ngog in a one-on-one with Kenny once more, with the Leeds keeper this time producing a great save.
Despite Wanderers being more threatening, the visitors looked dangerous on the counter, as Noel Hunt demonstrated when playing a smart one-two with Ross McCormack.
However, the former Reading man's shot fizzed past the wrong side of the left post.
Another breakaway move, this time set in motion by McCormack, saw Varney get the chance to double his tally but his right-footed shot just failed to curl into the net before half-time.
The Trotters then changed to a 4-4-2 formation when Beckford came on in place of captain Knight on the hour mark.
But it was the introduction of Rob Hall which almost saw Bolton get the chance to make their breakthrough. The former West Ham man went down in the box from a Stephen Warnock challenge, but referee Andy D'Urso waved play on.
Ngog then had another chance to equalise late on after he connected with Marc Tierney's cross. Once again, though, his shot went just wide and into the side-netting .
But the closest Wanderers came was in injury time as Pratley forced Kenny into making a tremendous save.
There was, however, still time for a late penalty shout, with Scott Wooton taking Pratley down in the box but once again it was not given, denying the Trotters the chance to take a much-needed point.