Striker Anthony Elding gave the visitors the perfect start when he netted his 21st of the season after just 10 minutes.
Strike partner Darryn Stamp threaded the ball into the path of the former Boston forward and he calmly rounded keeper Phil Smith to slot into an empty net.
Crawley had chances to hit back before the break through a Ben Judge volley which flashed just over and a 20-yard drive from Steve Burton which forced Alan Julian into a good save.
Boro made the hosts pay 11 minutes after the restart when captain Jason Goodliffe smashed a first-time shot into the roof of the net from eight yards following an Elding corner.
Crawley had calls for a penalty turned down on 75 minutes when a header by Ian Simpemba seemed to be handled on the line by former Reds defender Justin Gregory.
However, the home fans only had to wait a minute for Burton to pull one back when he fired under Julian from the edge of the box.
The hosts thought they had equalised 10 minutes from time when substitute Danny Davidson headed home a cross by Judge but the assistant referee ruled it out for offside.