Atleti's surprise home defeat to Levante a day earlier had given Ronald Koeman's side the chance to move six points behind Diego Simeone's league leaders.
They appeared to be on course to do just that after Lionel Messi marked a club-record 506th league appearance with a 32nd-minute penalty.
Alex Fernandez sealed a dramatic point for the visitors in the 89th minute, though, slamming home from the spot after Clement Lenglet was ruled to have clipped Ruben Sobrino.
The result ended Barca's seven-game winning streak in LaLiga and left them with 47 points after 23 games this season – their lowest tally at this stage of the season since the 2006-07 campaign when they amassed 46 points.
While Pique acknowledged it will not be easy to challenge Atleti for the title, he insisted they will not give up.
"It hurts a lot because Atletico's defeat gave us a chance," he told the club's official TV channel.
"Losing two points is difficult to take. Every game that you do not win, your options are reduced a little. There is an important distance [between Barca and Atleti], not unassailable, but it will depend on the next games. If we get good results, we will have options.
"It's difficult, obviously. We have to try. It's a hard blow. On Wednesday, we have another test against Elche.
"We have to recover our game and feelings because, if not, we could lose more games. We need to raise our heads, improve mentally and win against Elche."
Following the visit of Elche, Barca travel to Sevilla on Saturday.