Ospreys went home 24 - 20 winners but with no try bonus point, they will have to travel to play Munster in the play offs.
With other results in the PRO12, Pat Lam's men finish 7th in the league and will travel to play Gloucester next Sunday May 24th in the first of two play off games for Champions Cup rugby next year.
Hosts Connacht had the better of the first scrum between the two sides, but a mistake at the second exchange from Rodney Ah You gave Ospreys the penalty on halfway after five minutes.
Biggar, the GUINNESS PRO12's second highest points scorer prior to kick-off, stepped up and had both the distance and accuracy to put his side into the lead.
Connacht were causing some disruption to the Ospreys defence and Bundee Aki would have been in for the opening try if it wasn't for a good tackle to stop Aly Muldowney offloading in the corner.
Ospreys' early dominance in the scrum paved the way for the opening score though as they drove back the Connacht pack, turned over the ball and then Dan Baker set up Biggar to touch down under the posts.
The fly-half added the extras for a 10-0 lead and the visitors were soon paying another visit to the Connacht 22 as John added the Welsh region's second try of the afternoon on 26 minutes.
A penalty from Biggar gave the Ospreys a good platform in the corner and after a solid lineout the Ospreys backs opened up to allow centre John to run in - Biggar converting with ease.
A shell shocked Connacht side were being carved open and needed an excellent piece of defensive work from Jack Carty to deny Eli Walker a score.
Their defence was breached once again though just before the half-hour, this time the elusive Webb going over for his 11th GUINNESS PRO12 score of the season.
Just before the break Walker almost grabbed the bonus-point score but when reviewed by the TMO it was clear the winger had agonisingly knocked the ball on just over the try line.
A pair of early Carty penalties after the restart gave Connacht some much needed momentum and after earning another penalty within range of the posts a decision to go for the corner showed their intent.
The decision paid off as an excellent maul from the hosts, following a solid line-out, saw Eoghan Masterson crash over and after Carty added the extras it was a nine-point game all of a sudden.
Connacht were gunning for another score and got just that through prop Dennis Buckley in the final minute of action.
Carty added the extras for 24-20 and although there was time for a restart neither side were able to add any further score with both bound to be disappointed with the final score.