St Johnstone had enjoyed the better of the opening half hour with Alan Maybury hitting the post and Andy Jackson seeing a shot cleared off the line, but Chris Innes headed the visitors in front in the 36th minute.
Caley Thistle never looked in danger after Doran struck in the 57th minute and Richie Foran soon added a third as the Highlanders moved away from Hibernian.
Saints have now lost five games in succession without scoring and endured some bad luck in front of goal in the opening half, Murray Davidson threatening for the hosts with a deflected strike from 20 yards that went just wide.
Chris Millar curled over from a tight angle and Saints should have levelled before the break when Davidson flicked a long ball past the visiting defence before running clear, but the midfielder slipped the ball wide with only Ryan Esson to beat.
St Johnstone have scored just 18 goals in the SPL this season and they looked devoid of confidence as they sought a way back into the game, although Jackson curled just wide from 18 yards in the final phases.
Hamilton were denied their first Scottish Premier League home win of the season - and the chance to boost their slim survival hopes - when substitute Nick Blackman claimed a share of the points for Aberdeen in a 1-1 draw.
Aberdeen were awarded a penalty after 52 minutes when referee Calum Murray ruled that Steven Smith was fouled in the box by Gary McDonald. Chris Maguire stepped up for the spot-kick but was denied by a great save from Tomas Cerny.
Hamilton grabbed the opener after 64 minutes when Dougie Imrie - on his 50th appearance - raced onto a pass from Nigel Hasselbaink before keeping his cool and slotting past goalkeeper Jamie Langfield.
Langfield was then forced out of the action and replaced by Mark Howard just two minutes later after he was caught on the face by the boot of team-mate Zander Diamond.
Aberdeen replaced Scott Vernon with Blackman after 73 minutes and he grabbed the equaliser just two minutes later. Maguire supplied the pass, squaring across the face of goal for the on-loan Blackburn forward to rifle home at the far post.
Blackman was then denied his second when his shot bounced off Cerny and crashed off the upright as the match ended all-square.