The visitors looked good value for their lead as they had dominated the opening 20 minutes or so but Brakes began to turn the screw, particularly in the second half, but failed to extend visiting keeper Stuart Coburn until the game was into added on time.
Reeves bought the best out of Tony Breeden early on when he turned sharply and fired in a powerful effort that the Brakes keeper pushed round the post.
Leamington were struggling to get out of their own half for a time but went close when Jamie Towers nodded just past the post under pressure from Matt Doughty and Michael Williams after Liam Daly had knocked on a free kick from the right.
Matt Dodd was afforded very little space down the right wing, being very closely marshalled by the Alty defence, and when he was brought down on the right edge of the penalty area Coburn was off his line smartly to punch the free kick away from the advancing Daly.
The ball was played back into the penalty area, and although Stefan Moore did well to lift it over one defender it did not drop quickly enough for him to get a shot in on goal, and was cleared away.
The visitors moved possession out to the left and when Doughty’s cross was missed by Josh Green, Reeves was onto it in a flash, steering the ball low past Breeden to open the scoring.
Daly received a caution as he attempted to prevent the visitors from breaking clear again, Shaun Densmore firing the resulting free kick straight at Breeden.
James Walshaw was then substituted, looking very uncomfortable as he left the field having fallen awkwardly.
Reeves created the chance to fire in a second as he muscled his way past Daly inside the area, but Breeden was equal to his shot this time, saving well.
Leamington then began to work their way back into the game.
The visitors gave away a corner from which Daly hit a close range drive at the far post that deflected off a defender and struck Elliot Sandy on the back before being cleared away.
Lee Chilton slung the ball back in from the left, but Coburn reacted quickly to claim the ball as Daly knocked it first time back into the goalmouth.
Nicky Clee was booked for manhandling Dodd to the turf on the right, and from the free kick Daly’s powerful header was helped behind for one of a succession of corners Brakes forced during the first half, but they counted for little.
The pressure became more intense as the second half wore on.
Chilton saw a shot blocked on the edge of the area, and his ball back into the box after Towers had burst down the right and clipped it into the middle was struck into a defender by Matt Dodd, the vague shouts for a penalty being waved away by the referee.
Altrincham were well organised defensively and continued to work hard, restricting Leamington’s movement around their penalty area. They briefly threatened at the other end, Daly being forced to head a dangerous cross behind, and Reeves fired a first time volley wide from a free kick.
Alty were also shouting for a penalty after a coming together inside the area, but again there was nothing forthcoming from the officials.
As the game moved into injury time Brakes inevitably had one more chance to snatch something.
Chilton emerged with the ball from a scramble on the edge of the box, turned inside a defender and fired a low shot across goal, only for Stuart Coburn to get a vital touch to the ball, just preventing Stefan Moore from being able to force it home at the far post.
Daly headed just over from the corner but Altrincham held on for the points.
n Brakes’ next game is a trip to Stockport County tomorrow (Saturday, ko 3pm).Like Brakes, Stockport sit just above the bottom three with one win and a draw so far this season.