Manchester City's quadruple bid was shattered by a "cruel" Champions League quarter-final exit on Wednesday.
Arsenal held out for a first clean sheet away from home in the Premier League this season to move ahead of Chelsea on goal difference into fourth.
A much-needed win moved Ole Gunnar Solskjaer's men up to fifth.
The decisive second leg will be played at Manchester City's Etihad Stadium on April 17.
Manchester City's 2-0 victory over Cardiff took Pep Guardiola's quadruple-chasers to 80 points - one clear of Jurgen Klopp's Liverpool.
Arsenal climbed two points above fourth-placed Tottenham and fifth-placed Manchester United to bolster their chance of qualifying for the Champions League.
The Premier League title race is set to go down to the wire and the battle for top-four places is tighter than ever.
Mauricio Pochettino's men threw away an early lead through Harry Kane to lose 2-1 at Southampton.
This is the first time Tottenham Hotspur are in the last eight of the Champions League since 2010/11.
Harry Kane's contentious penalty 16 minutes from time ensured Tottenham maintain a four-point lead over Arsenal in the fight for top-four finish.
Jurgen Klopp's men calmed fears over their staying power following a recent dip in form as they chase down their first top-flight title since 1990.
Maurizio Sarri, who reacted furiously when Kepa Arrizabalaga refused to be substituted, said the incident had been a "misunderstanding".
The fall-out from Maurizio Sarri's astonishing row with Kepa Arrizabalaga threatens to cause further damage in a crucial London derby against Tottenham.
Mauricio Pochettino's side would have moved within two points of leaders Manchester City with a win at Turf Moor.
Tottenham Hotspur would have moved within two points of leaders Manchester City with a win at Turf Moor.
Mauricio Pochettino said that Jadon Sancho had been on his radar for a while before he started making headlines.
Manchester City demolished Chelsea 6-0 to go top while Tottenham Hotspur cemented their third place with a 3-1 win over Leicester City.