Jan Vennegoor of Hesselink and Barry Robson scored before the interval for the Bhoys with Lionel Messi grabbing Barca's sole response.
The visitors equalised for a second time thanks to former Arsenal star Thierry Henry before Messi slotted home a 79th-minute winner.
Celtic manager Gordon Strachan caused a shock in his starting line-up by bringing Paul Caddis in for the cup-tied Andreas Hinkel, while Robson replaced the suspended Scott Brown in midfield.
Celtic keeper Artur Boruc was forced to make a smart save to deny Andres Iniesta, before Caddis headed Henry's shot clear.
However, the hosts took the lead with their first attack of the match. Scott McDonald found Lee Naylor who crossed for Hesselink to power home a header.
Almost immediately Barcelona were level though when Messi beat Naylor and, after playing a one-two with Deco, fired a superb shot into the roof of the net.
Ronaldinho went close to giving Frank Rijkaard's side the lead with a free-kick before Celtic regained their advantage when Aiden McGeady crossed for Robson to head over Barca keeper Victor Valdes.
After the interval though, the Catalan giants took control with Ronaldinho setting up Henry to grab a sublime equaliser before Messi beat Gary Caldwell and fired a shot past Boruc.
Barcelona's second-half comeback leaves Strachan's side with a mountain to climb in the second leg of their last 16 encounter in the Camp Nou in two weeks' time.