Lowly Granada were organised and dangerous on the break and it was only after Victor Valdes had saved a one-on-one with Fabian Orellana in the 86th minute that Barca finally found a way past inspired visiting goalkeeper Tono.
Xavi cracked a shot in off the underside of the bar in the 87th and Messi's persistence saw him squirm into the area and strike a low cross that was turned into his own net by Borja Gomez in time added on.
"Every victory is important. Maybe we struggled more than on other days but we have to know how to get through these games," Barca full-back Dani Alves told Spanish television.
"It wasn't that we didn't play well, a lot of credit must go to our rivals. Their keeper has been superb and I have to congratulate him."
The pressure grew on the visitors' goal with Messi, Cesc Fabregas and David Villa, on his first start since breaking his leg last December, all going close.
Tono's work load was increasing when Granada's Orellana broke free to be denied by Valdes, and soon after Xavi struck the opener from the edge of the area.
The Brazilian defender connected with a free-kick after 18 minutes but his side failed to make the most of a second yellow card for Espanyol's Wakaso Mubarak in the 68th, and endured some worrying moments before securing the victory.
Celta's second win put them ninth in the standings with six points from five games, two places ahead of Real Zaragoza, who won 3-1 at home to bottom side Osasuna.
