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Chasing 161, Royals rode on some brilliant contribution from their top order as they crossed the line with five balls to spare.
Speaking at the post-match presentation ceremony, Smith said: "It was a good end to our home season as the boys pulled things back nicely."
Royals won only three out of their seven matches they played at home at the Sawai Mansingh Stadium here in the ongoing IPL edition.
Smith also praised Liam Livingstone (44) and Sanju Samson (48 not out) for their efforts with the bat.
"Liam Livingstone opening for the first time in IPL hit the ball cleanly and Sanju finished it off well. We're playing together as a group and won those key moments. At the start we weren't doing it. We've had a lot of close games and fortunately last few we've been able to get over the line," he said.
Earlier, SRH looked on course of a big total with opener David Warner (37 off 32) and Manish Pandey (61 off 36) putting up a 75-run stand for the second wicket. However, Jaydev Unadkat-inspired Rajasthan chipped in with wickets at crucial juncture and restricted Hyderabad to 160-8.
With Saturday's win, Royals have theoretically kept alive their playoff chances in the ongoing IPL, having 10 points from 12 games.
To stay in contention for making it to the playoffs, Rajasthan must win their remaining two away games against Royal Challengers Bangalore (RCB) and Delhi Capitals and then hope for favourable results from other matches.
"We need to continue winning and you never know. I have got one more to play in Bangalore, so it'll be nice to finish on a winning note. Hopefully the boys will play well in Delhi as well," said Smith who will be heading to Australia after the game against RCB on Tuesday.
Unadkat, who was adjudged the 'Player of the Match' for taking two wickets and three catches, said he needed this performance to boost his confidence. "I have had a tough time and I needed this to boost my own confidence. The management and Smudge (Smith) have been backing me. I needed them in such tough moments and I am really glad."
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