Manchester City clinched a 1-0 victory at Tottenham Hotspur, while Chelsea were held to a 0-0 draw at home by Aston Villa.
It is their best finish for 54 years, since the glory days of the 1960s when the club was setting new standards in English football under Bill Nicholson.
Harry Kane's hat-trick turned Tottenham Hotspur's potentially tricky FA Cup fifth-round tie at Championship side Fulham into a Sunday stroll by the River Thames as they won 3-0 at Craven Cottage.
A Robbie Keane penalty gave Tottenham Hotspur a 1-0 victory over Reading on Sunday to move the London club up to sixth in the Premier League.
Images from the English Premier League matches played on Sunday
Tottenham Hotspur endured a frustrating time in front of goal on Saturday as West Ham United held the Champions League quarter-finalists 0-0 in an entertaining Premier League game at White Hart Lane.
Tottenham Hotspur brutally exposed Liverpool's defensive frailties as Harry Kane plundered two more goals in his side's 4-1 romp at Wembley on Sunday.
The 56-year-old Portuguese returns to management for the first time since being sacked by Manchester United in December, and replaces Mauricio Pochettino, who was sacked on Tuesday.
Tottenham Hotspur have been fined 15,000 euros ($18,600) by UEFA after their Europa League game against Partizan Belgrade last month was suspended for several minutes following three separate pitch invasions, the governing body said on Thursday.
A summary of Saturday's action in the Premier League.
Tottenham Hotspur guaranteed a top-four finish in the Premier League when Peter Crouch's late header secured a gutsy 1-0 win at Manchester City on Wednesday.
IMAGES from the EPL matches played on Saturday.
Tottenham Hotspur and Newcastle United both sealed comeback victories to boost their European qualification hopes.
Christian Eriksen kept Tottenham Hotspur's hopes of completing a remarkable salvage job and reaching the last 16 of the Champions League alive with a late winner in his side's 1-0 victory over Inter Milan at Wembley on Wednesday.
Brief reports of English Premier League matches on Saturday:
Andre Villas-Boas has signed a three-year contract as manager of Tottenham Hotspur, the Premier League club said on Tuesday.
Tottenham Hotspur manager Mauricio Pochettino has agreed a two-year contract extension that will keep him at White Hart Lane until 2021.
Arsenal had more than twice as many chances but failed to take any as the visitors defended gamely.
The Premier League club did not say who would replace Jol, although Seville's Juande Ramos is expected to take over.
Tottenham Hotspur manager Harry Redknapp aims to finish the season with the Premier League club despite the clamour for him to be appointed as England coach, he said.
Aston Villa kept Tottenham Hotspur bottom of the Premier League on Monday with a deserved 2-1 victory at White Hart Lane. Villa scored after five minutes through Nigel Reo-Coker and again early in the second half when Ashley Young's shot squirmed under Tottenham's Brazilian goalkeeper Heurelho Gomes.
Man City to face Spurs in League Cup
Tottenham Hotspur steamed into the top four with Harry Kane marking his 100th Premier League appearance with a double in a 4-1 thrashing of Watford on Sunday.
Tottenham Hotspur took a large bite out of Leicester City's Premier League lead with two goals apiece from Harry Kane and Dele Alli earning them a 4-0 victory at Stoke City on Monday.
Tottenham Hotspur, one of the country's best-known soccer clubs, is to go private, saying the cost of a stock market listing would be better spent helping expand its stadium.
Manchester United's Alejandro Garnacho scored with a magical overhead kick in the third minute to send the visitors on their way to a 3-0 victory over an Everton team already reeling in their first Premier League game after being docked 10 points.
News of all that's transpired on and off the football field
Egypt striker Mido has returned to Tottenham Hotspur after the club reached agreement on a deal with AS Roma.
Tottenham Hotspur just beat the transfer window deadline to sign midfielder Danny Murphy from Charlton Athletic.
Chelsea ticked off three more points in their Premier League title quest with a 2-1 victory over Manchester City on Wednesday but Tottenham Hotspur's Houdini-like 3-1 win at Swansea means the race is far from over.
Reigning champions City are also involved in a separate independent hearing into the club's alleged breaches of Premier League financial regulations which began in September, with a verdict expected some time before the end of the current season.
A summary of Sunday's action in the Premier League.
'It's alleged that, during the 94th minute of the fixture, the club failed to ensure their players do not behave in way which is improper.'
Tottenham Hotspur manager Harry Redknapp has said he is "definitely interested" in bringing LA Galaxy's former England captain David Beckham to the Premier League club on loan this month.
Bukayo Saka scored either side of halftime as Arsenal crushed second-bottom Burnley 5-0.
A virtuoso performance from Sergio Aguero earned Manchester City a 4-1 victory over Tottenham Hotspur in the Premier League on Saturday, the Argentine scoring all his side's goals in an incident-packed clash at The Etihad.
Tottenham Hotspur's hopes of a first victory over Chelsea at Stamford Bridge in 26 years disappeared as a 2-2 draw left Leicester City celebrating the Premier League title on Monday.
Tottenham Hotspur could face punishment from European soccer's governing body UEFA after their Europa League game with Partizan Belgrade on Thursday was suspended for several minutes following three separate pitch invasions.
Tottenham Hotspur hammered 10-man Aston Villa 4-0 in the Premier League on Sunday thanks to four second-half goals, allowing them to close the gap with their Birmingham rivals to two points in the chase for Champions League football next season.
