Karlan Grant halved the deficit after 35 minutes, turning his marker to net with a right-footed shot from an angle, but seconds later Eliasson headed home a Brownhill cross.
Eliasson was causing havoc and his 40th-minute cross was converted by Ashley Williams with a bullet header at the near post.
Four minutes after the break it was 5-1 as Weimann ran onto an Adam Nagy through ball and rounded goalkeeper Grabara before firing home from a narrow angle.
Juninho Bacuna pulled a goal back with a sweet low drive from 25 yards, but it proved no more than a consolation for his side despite their hard-fought performance in the second half.
Huddersfield suffered a blow after only five minutes when striker Fraizer Campbell hobbled off to be replaced by Adama Diakhaby. Grant moved up front from the left flank and caused plenty of problems apart from his goal in an open first half.
His 18th-minute volley from 25 yards brought a brilliant fingertip save from Robins goalkeeper Dan Bentley as the visitors recovered from an uncertain start that saw the hosts go in front.
Grabara did equally well to save Eliasson's close-range shot after a superb 22nd-minute through ball from Nagy had put the winger in on goal.
Huddersfield looked well in the game when responding to Kongolo's own goal with Grant's strike, but inability to defend crosses cost them dear as Eliasson and Williams put City out of sight by the break.
Bacuna's goal sparked a brief spell of pressure but – at the other end – Grabara had to produce a good save to keep out a Tommy Rowe shot.
Neither Huddersfield's players, nor their 800 travelling fans ever gave up and Cowley's men remained a threat going forward right to the death.
But they could not match City's cutting edge on a day when Lee Johnson's team produced some scintillating attacking football to achieve a result that saw them move into the play-off zone.