Grabban slotted past Rob Green on 59 minutes for his 20th goal of the season to put the home side back in front after Armand Traore had brilliantly cancelled out Tommy Elphick's opener.
Despite having Harry Arter sent off on 67 minutes, the Cherries' fourth win in a row - alongside a Derby defeat at Middlesbrough - secured the Foxes' place back in the top flight.
Redknapp began his managerial career on the south coast in 1983, spending nine years in charge of the club.
Despite the hype of the former Tottenham boss bringing a team competitively to the Goldsands Stadium for the first time in 22 years, the game started slowly.
It took until the 15th minute for any action, with former England international Jermaine Jenas latching onto a ball from Junior Hoilett to send a stinging shot towards the top corner but Bournemough goalkeeper Lee Camp made a diving save.
Ravel Morrison was the next to try his luck after the lively Hoilett picked him out just inside the box, but Camp was on hand to tip his bouncing effort over the bar.
The near-misses seemed to wake Eddie Howe's top-six chasers up, with Matt Ritchie sending a shot wide and Marc Pugh seeing his goalbound shot blocked by Danny Simpson.
Bournemouth kept attacking and should have opened the scoring on 36 minutes when Grabban squared for Yann Kermorgant, but Green made a fantastic point-blank save.
The honour of the first goal went to Elphick early in first-half stoppage time with the captain nodding home a pinpoint Ian Harte corner.
Redknapp must have read the riot act in the interval, though, as seconds after the restart his side were level through Traore's fine strike.
Tom Carroll lofted a delightful crossfield ball into the path of the Senegalese, who juggled the ball to the byline before smartly firing past Camp.
If the Rangers fans thought the goal would inspire a much-needed three points, though, they were wrong.
After Grabban's goal, Redknapp turned to star striker Charlie Austin in a bid to get back into the game and his job was made easier on 67 minutes when Arter got his marching orders from referee Jonathan Moss for a high tackle on Hoilett.
Despite having a one-man advantage and going close through Hoilett, the London outfit would have gone further behind but for Green making solid saves from Andrew Surman, Grabban and Harte.
Richard Dunne almost snatched a late leveller, but saw his header cleared off the line in the final seconds.