Valverde was eager for a swift return to winning ways following the criticism which greeted Saturday's surprise LaLiga loss at Levante.
But the anticipated home win failed to materialise as below-par Barca produced an uninspiring performance against determined opposition.
Lionel Messi was denied a superb first-half goal by the crossbar, while both sides had efforts disallowed for offside at the Nou Camp.
The Spanish side retain top spot in Group F but with qualification far from assured ahead of difficult games against Borussia Dortmund and Inter Milan.
Valverde's side had struggled to overcome Slavia during their 2-1 win in Prague a fortnight ago and again endured further difficulties in front of their own fans.
The adventurous visitors almost opened the scoring in the 27th minute when Nicolae Stanciu was picked out by Peter Olayinka's pull back but saw his goal-bound shot strike a team-mate and fly over the crossbar.
Luis Suarez's calf injury had resulted in a rare opportunity for Barcelona forward Ousmane Dembele.
France international Dembele, making only his third start of the season, was selected alongside Messi and Antoine Griezmann for the first time, but the hosts' front three initially found chances at a premium.
Captain Messi, so often the catalyst for Barca's success, sought to change that and almost conjured up a sensational opener 10 minutes before the break.
The Argentinian collected the ball on the halfway line and accelerated away from Slavia defender Michal Frydrych before cutting inside and smashing the woodwork from the edge of the box.
Messi, searching for his 114th Champions League goal, was then denied by Ondrej Kolar from a tight angle, before the Slavia goalkeeper kept out Gerard Pique's thumping header from the resulting corner.
Barcelona last lost a Champions League group game on home soil more than 10 years ago, when Russian club Rubin Kazan secured a shock 2-1 success against Pep Guardiola's side in October 2009.
That record looked in considerable danger as the away side continued to cause problems.
Jan Boril, who scored Slavia's goal against Barca in the Czech capital, thought he had repeated the trick in first-half stoppage time before his finish on the rebound was chalked off by an offside flag.
Valverde, whose side had been on a seven-match winning run before their weekend loss, desperately needed a vastly improved second-half performance but it was not forthcoming.
Griezmann, who has managed only four goals since his big-money summer switch from Atletico Madrid, was extremely ineffectual, leaving Messi to try and force a breakthrough.
Keeper Kolar saved his best until last, repelling Messi's close-range effort 13 minutes from time as Slavia held on for a famous point which was celebrated exuberantly by the travelling fans and dents Barca's qualification hopes.