Striker Barnett scored twice in the 4-2 win over Cheltenham on Tuesday night.
Sanchez Watt, on loan from Arsenal, also scored two goals to lift Crawley, who have lost only one of their last 23 games in all competitions, above Cheltenham to the top of League Two.
Barnett, who now has 15 goals this season, rejected a £900,000 move to Blackpool during the January transfer window after failing to agree personal terms.
Crawley boss Steve Evans said: "I'm sure we'll get a few more phone calls now.
"We had West Ham here and Peterborough, a host of top Championship clubs are interested, and his first goal was simply the best I've seen.
"If it had been scored in a Premier League match you would have seen it replayed over and over. It was sensational."
Luke Garbutt had given Cheltenham an early lead but Watt and Barnett took over before Darryl Duffy grabbed a late consolation.
