Connacht 19 Glasgow 19
Glasgow missed their chance at a rare win as a try from Glasgow in the game's last play forced a draw at the Sportsground.
An injury-time try from flanker James Eddie saw the Warriors dash Connacht's hopes of a much-needed win and, although replacement Colin Gregor flashed his touchline conversion wide of the uprights, Glasgow were still the happier side at the final whistle.
Leading 14-3 thanks to three Ruaridh Jackson penalties and a try from DTH van der Merwe, Glasgow looked set to take the spoils.
However, bottom side Connacht engineered a tremendous fightback, with Ian Keatley kicking two penalties and converting a slick solo try from Gavin Duffy.
Keatley's fourth successful penalty, three minutes from time and with Glasgow's Calum Forrester in the sin-bin, looked to be the match-winning score, but Eddie had other ideas and he scrambled over in the left corner, with the grounding confirmed by TMO Dermot Moloney.
Connacht showed their intent early on with a barnstorming run from George Naoupu and chip through from Fionn Carr, but Glasgow were able to deal with the danger.
Warriors' Argentinian star Bernardo Stortoni threatened in broken play and the visitors built forward pressure before Jackson sent a tame drop-goal effort wide of the posts.
But Glasgow were first off the mark after 12 minutes when Connacht prop Brett Wilkinson was pinged for a ruck and Jackson nailed a kick from a central position.
Jon Welsh and Richie Vernon carried well for Glasgow as they tried to tie the Connacht defence in, but a block down by hooker Sean Cronin got the Irish province on the march and their first bit of decent territory yielded a successful penalty for Keatley, 24 minutes in.
That good work was undone, three minutes later, when Johnny O'Connor was singled out for going off his feet at a ruck and Jackson quickly made it 6-3.
The first 40 minutes had been pockmarked with mistakes, and a Connacht knock-on and offside, as the second half began, allowed Jackson to kick Glasgow 9-3 clear.
Jackson then missed his first place-kick, after Connacht lock Mike McCarthy saw yellow for infringing at a ruck, and Carr showed his attacking skills, hacking on twice before van der Merwe rescued the situation for the Scots.
Glasgow put 11 points between the sides in the 48th minute when Jackson initiated a purposeful counter-attack and big Canadian winger van der Merwe finished off a slick move down the left involving four players.
But the try went unconverted and Michael Bradley's men managed to play their way back into contention.
Carr, Aidan Wynne and the hard-working Naoupu took the ball up and Glasgow were marched back into their 22 before full-back Duffy ran against the grain and caught Glasgow on the drift as he eased past tackles from Mark McMillan and Ed Kalman to touch down behind the posts.
Keatley tapped over the conversion and maintained his 100% record with a fourth penalty but 14-man Glasgow increased the tempo with a series of late pick and drives, sucking a tiring Connacht defence in before Eddie found space to squeeze over in the corner.