Dannie Bulman cancelled out Paul Hakim's first-half goal shortly after the break and Jake Edwards sealed the points six minutes from time.
Crawley's new management team, who took over following the sacking of John Hollins on Monday, were dealt a blow before kick-off with the news that two players had also left the club.
Former club captain, Danny Brown and squad player Pat Sattleton, who are both on long contract terms, walked out on Friday.
It left Reds with only 11 fit players and a bench full of teenagers and it was little surprise when Saints dominated the early stages.
The visitors should have gone ahead after just four minutes when Hakim beat the offside trap to go clear on goal.
But he decided to try to lob Ben Hamer and miss-hit his efforts straight at the keeper.
Striker Shaun Batt, who was making his debut, then twice went close, first when he failed to get a connection on a dangerous low cross from Matt Hann and then when he was denied by the smothering Hamer following a mazy dribble.
Having weathered the earlier pressure, Crawley gradually improved and had a golden chance on 22 minutes.
Woozley was left all alone in the six-yard box at a Lee Blackburn free kick and somehow managed to guide a volley wide.
The miss proved to be costly as Hakim gave Saints the lead five minutes later with his eighth goal of the season.
Winger Ben Walshe gained possession on the left flank following a poor clearance by Woozley and picked out the unmarked Hakim who nodded into the bottom corner with ease.
The home side responded well to the setback and threatened three times before half time through Tony Scully.
He first had a volley well saved by keeper Ricky Perks, curled just wide after a neat exchange of passes with Edwards moments later, before chipping just past the post following a clever lay-off by Scott Rendell.
Reds remained in control after the break and had two good opportunities minutes into the re-start.
Woozley headed over from a Scully cross and Rendell fired into the side netting with an angled drive from the edge of the box.
Crawley got the equaliser they deserved on 56 minutes, thanks to the impressive Scully.
He played a superb one-two with Edwards and crossed from the right wing for the on-rushing Bulman to drill home from six yards.
Scully, the best player on the pitch by a long way, sent panic into the St.Albans defence again soon after with another run which ended in a shot that sped inches wide.
The hosts lay siege on the Saints goal for the rest of the game and wasted numerous good chances before finally getting their second.
Rendell dragged a shot the wrong side of an upright after being put through by Edwards.
Scott Hiley then hit the inside of a post with a curling shot from the inside left channel moments later.
Gary Mills and Blackburn both had long-range efforts blocked bravely by defenders as the visitors hung on.
But their resistance was broken on 84 minutes following a triple substitution at a corner by St. Albans boss, Colin Lippiatt.
Edwards was not marked properly eight yards from goal and rose highest to power a header past Perks from Scully's flag-kick.
But the Saints almost snatched an undeserved point in the dying seconds when Dave Theobald had a header from a corner scrambled off the line.
Crawley's new management team, who took over following the sacking of John Hollins on Monday, were dealt a blow before kick-off with the news that two players had also left the club.
Former club captain, Danny Brown and squad player Pat Sattleton, who are both on long contract terms, walked out on Friday.
It left Reds with only 11 fit players and a bench full of teenagers and it was little surprise when Saints dominated the early stages.
The visitors should have gone ahead after just four minutes when Hakim beat the offside trap to go clear on goal.
But he decided to try to lob Ben Hamer and miss-hit his efforts straight at the keeper.
Striker Shaun Batt, who was making his debut, then twice went close, first when he failed to get a connection on a dangerous low cross from Matt Hann and then when he was denied by the smothering Hamer following a mazy dribble.
Having weathered the earlier pressure, Crawley gradually improved and had a golden chance on 22 minutes.
Woozley was left all alone in the six-yard box at a Lee Blackburn free kick and somehow managed to guide a volley wide.
The miss proved to be costly as Hakim gave Saints the lead five minutes later with his eighth goal of the season.
Winger Ben Walshe gained possession on the left flank following a poor clearance by Woozley and picked out the unmarked Hakim who nodded into the bottom corner with ease.
The home side responded well to the setback and threatened three times before half time through Tony Scully.
He first had a volley well saved by keeper Ricky Perks, curled just wide after a neat exchange of passes with Edwards moments later, before chipping just past the post following a clever lay-off by Scott Rendell.
Reds remained in control after the break and had two good opportunities minutes into the re-start.
Woozley headed over from a Scully cross and Rendell fired into the side netting with an angled drive from the edge of the box.
Crawley got the equaliser they deserved on 56 minutes, thanks to the impressive Scully.
He played a superb one-two with Edwards and crossed from the right wing for the on-rushing Bulman to drill home from six yards.
Scully, the best player on the pitch by a long way, sent panic into the St.Albans defence again soon after with another run which ended in a shot that sped inches wide.
The hosts lay siege on the Saints goal for the rest of the game and wasted numerous good chances before finally getting their second.
Rendell dragged a shot the wrong side of an upright after being put through by Edwards.
Scott Hiley then hit the inside of a post with a curling shot from the inside left channel moments later.
Gary Mills and Blackburn both had long-range efforts blocked bravely by defenders as the visitors hung on.
But their resistance was broken on 84 minutes following a triple substitution at a corner by St. Albans boss, Colin Lippiatt.
Edwards was not marked properly eight yards from goal and rose highest to power a header past Perks from Scully's flag-kick.
But the Saints almost snatched an undeserved point in the dying seconds when Dave Theobald had a header from a corner scrambled off the line.