Liverpool kept alive their hopes of qualifying for next season’s Champions League with a priceless 2-1 victory over Everton in the Merseyside derby at Anfield on Sunday.Liverpool led through first-half goals from captain Steven Gerrard and Spanish playmaker Luis Garcia before a 77th minute red card for their Czech striker Milan Baros threw Everton a lifeline. Everton’s Australian midielder Tim Cahill capitalised with a fine strike on 82 minutes but Liverpool held on during a thrilling finale for a vital win in the race to finish fourth.Birmingham City beat Aston Villa 2-0 in the Midlands derby and two Peter Crouch goals hauled Southampton out of the Premier League relegation zone in a 3-1 win at Middlesbrough in the day’s other games.