Crystal Palace moved six points clear of the relegation zone with victory over Brighton, Huddersfield also boosted their survival chances with a 1-0 win against Watford and Everton were held to a 1-1 draw by Swansea at the Liberty Stadium.
Mark Hughes looked set for a first league win as Southampton manager as they led the reigning champions 2-0 through Dusan Tadic and debutant Jan Bednarek with just 20 minutes remaining.
The Seagulls nearly came back from being dead and buried after finding themselves two down at Palace after just 14 minutes.
"As long as it is mathematically possible we will believe we can reach the top four", added Giroud.
Liverpool proved too strong for Bournemouth after clinching a 3-0 win at Anfield.
Crystal Palace had been set to drop into the bottom three on goal difference prior to Chelsea's comeback.
Olivier Giroud scored twice as Chelsea came from 2-0 down to win 3-2 and leave Saints five points from safety.
Wilfried Zaha scored twice for the hosts as all five goals arrived before half-time.
All five goals from both sides were scored in the first half of the game, with Roy Hodgson's side edging out their fierce rivals to seal all three points.
Tom Ince scored a 91st-minute victor to help Huddersfield beat Watford 1-0.
Ayew smartly got onto the end of a Tom Carroll ball to beat Everton goalkeeper Jordan Pickford after Kyle Naughton's own goal handed the Toffees the lead in the first half.
The Clarets are only two points behind sixth-placed Arsenal, who face Newcastle at 13:30 BST on Sunday.
Zaha restored the two-goal advantage with a bullet header from Milivojevic's flawless cross and, in a frenetic opening 45 minutes, Jose Izquierdo struck with a fine finish for the visitors.