Trailing 3-2 entering the seventh inning, the O's exploded for five runs off Rays reliever Jake McGee (0-1) to take the lead.
Jones' double erased Baltimore's one-run deficit and David capped the outburst with his blast as Baltimore banged out 13 hits en route to its first victory of the season.
Jones finished 3-for-5 with two runs scored while Matt Wieters cracked a two-run homer in the victory.
Jason Hammel (1-0) earned the win in his first career Opening Day assignment, surrendering three runs on just three hits over six innings.
Rays starter David Price got a no-decision after limiting Baltimore to two runs over six innings.
Ben Zobrist homered and drove in two runs for Tampa Bay.