Oduro scored his fourth goal of the season in the 32nd minute, while Lekic tied it in the 48th minute with his second of the season.
Both clubs found numerous opportunities to get the winner late, but goalkeepers Sean Johnson and Matt Reis stopped the remaining shots they faced.
Ray Guy, making his Revolution debut, came up with the defensive effort of the night when he cleared an Orr Barouch ball off the line in the 73rd minute.
With the draw, the Fire improved its record to 2-4-9, while the Revolution extended its winless streak to six straight and had its record go to 3-7-6.