Costa set up Eden Hazard's early opener, which was cancelled out by Oriol Romeu, before Gary Cahill restored the hosts' lead at an unseasonably cold Stamford Bridge.
Subrata Paul, from West Bengal, has been one of India's finest goalkeepers. He was the number one goalkeeper during the time Englishman Bob Houghton was at the helm of affairs as India coach and Bhaichung Bhutia was the captain.
Lionel Messi had broken a three-year goalless drought against Madrid to cancel out Casemiro's opener for Madrid in the first-half.
Arsenal's third FA Cup final in the last four years -- they won it in 2014 and 2015 -- gives them a record 20 appearances in the showpiece match of the famous old competition.
Nemanja Matic's thunderous long-range effort sealed a final date for Chelsea against Arsenal or Manchester City, who meet in the second FA Cup semi-final on Sunday.
Holders Real Madrid face Atletico Madrid in the Champions League semi-finals as the city rivals do battle once more and French surprise-package Monaco take on Juventus, following the draw on Friday.
The scrappy, sapping victory means Jose Mourinho's Manchester United remain on course to claim the only major trophy missing from the club's collection.
Juventus join holders Real Madrid, Atletico Madrid and Monaco in Friday's semi-final Champions League draw.
Jamie Vardy scored in vain as Leicester City's Champions League fairytale came to a valiant end following a 1-1 quarter-final draw at home to Atletico Madrid on Tuesday.
Cristiano Ronaldo scored a controversial hat-trick as Real Madrid overcame 10-man Bayern Munich 4-2 after extra time in Tuesday's thrilling quarter-final the second leg.
Cristiano Ronaldo made history by becoming the first player to reach 100 Champions League goals as he scored a hat-trick in Tuesday's quarter-final second leg against Bayern Munich.
Goals from Alexis Sanchez and Mesut Ozil ended Arsenal's run of four successive away defeats with a scratchy 2-1 victory over struggling Middlesbrough in their Premier League clash on Monday.
Despite being Chelsea's captain, John Terry has become an increasingly peripheral figure under Italian manager Antonio Conte and has made just five league appearances so far this season.
After appearing set to cruise to the title for most of the season, Chelsea now face a nerve-jangling final six matches as they try to hold off a strong challenge from Tottenham Hotspur.
Vincent Kompany marked his return to the Manchester City line-up with a rare goal as his side climbed to third in the Premier League with a 3-0 win at Southampton on Saturday.
Vincent Janssen bagged a rare goal in second-half stoppage time as Tottenham Hotspur chalked up seven successive league wins for the first time since 1967.
Leander Dendoncker struck late to snatch a 1-1 draw for Anderlecht against Manchester United in the first leg of Thursday's Europa League quarter-final, while crowd trouble overshadowed Lyon's tie with Besiktas.
In the other two matches, Atletico Madrid defeated Leicester City 1-0 and Monaco beat Borussia Dortmund 3-2 in a thriller.