Poland international Rasiak has only notched four goals at home out of 18 in total this season and has not scored at St Mary's since the Birmingham City game on November 25th.
Rasiak missed his first opportunity in the 18th minute. Gareth Bale put over a pinpoint cross from the left-hand side, but Rasiak could only guide his free header three yards wide of the left-hand post.
Rasiak's second golden opportunity came when Bale picked him out on the far post from a corner, but he headed wide again and should have scored.
Burnley have not won for eight games, stretching back to November 28th, and seemed happy to settle for a point. Ade Akinbiyi made his first league start in his second spell at the Lancashire club since his £750,000 move back from Sheffield United.
He toiled away upfront all afternoon, but had little to show for his efforts.
Burnley's other new signing Eric Djemba-Djemba was the first to threaten on six minutes when he fired a right-foot shot over the bar from 20 yards.
Southampton's starlet Bale underlined why so many Premiership clubs are chasing him with a series of well-taken free-kicks.
In the eighth minute Bale sent a free-kick from 20 yards fizzing over the Burnley bar with debutant keeper Michael Pollitt well beaten.
On the half-hour mark, Kenwyne Jones took the ball past Jon Harley and centred, but John McGreal cleared the danger with Rasiak waiting to pounce.
Harley was booked for bringing down Mario Licka and Southampton defender Darren Powell followed him into the book for pushing.
Southampton midfielder Pele fired a right-foot shot from 25 yards just over the bar.
Djemba-Djemba had the last say in the first half when he put a free-kick in injury time just over the bar.
Burnley were missing ten-goal top scorer Andy Gray and were lacking a real cutting edge upfront.
Kyle Lafferty threatened for the Clarets on 51 minutes but could only put his right-footed shot straight at Kelvin Davis in the Saints goal.
Bale had two other free-kicks, on 56 and 68 minutes, but both of them flew just over the bar.
Rasiak hit a right-foot lob from 20 yards just wide of the right-hand post on 79 minutes as Southampton desperately tried to find the winning goal.
Licka hit a 30-yard left-foot shot that brought out of Pollitt his first meaningful save of the game in the 82nd minute.
Southampton continued to press for the winner as Burnley sat back and settled for a point, but the home side could not break the Clarets defence down.