Fly-half O'Connor, who was off target twice in the first half, made amends with the last kick of the game after Tiernan O'Halloran had given Connacht hope with a try midway through the second period.
Glasgow, who have lost only two RaboDirect PRO12 games since September, were comfortably in front with 25 minutes remaining thanks to the boot of Ruaridh Jackson and a first-half Jon Welsh try but a late onslaught from the Connacht pack had the desired effect.
The Irish province had not played at Galway Sportsground since their 15-13 New Year's Day defeat to Leinster and it showed in a scrappy opening, with handling errors costing both sides early scoring opportunities.
That error-prone start was highlighted when O'Connor missed two early penalties for the hosts and frustrations amongst the visitors were heightened when Warriors scrum-half Henry Pyrgos miscued his attempted drop goal.
Thankfully, Jon Welsh took matters into his own hands on 26 minutes to barge under the posts for his side's only try. Jackson coolly added the extras.
As the half was drawing to a close, O'Connor made amends for his earlier misses by slotting over three points to reduce Connacht's half-time arrears to 7-3.
That kick was supposed to galvanise struggling Connacht, who instead gifted their Scottish opponents an easy three points that Jackson gleefully took.
Samoan winger David Lemi attempted to increase Warriors' advantage with an ambitious drop goal which drifted well wide, and the Irish province sensed their opportunity as the game sailed past the hour mark.
Connacht's pack took hold and a drive all the way to the Glasgow try-line brought a penalty on 64 minutes.
Instead of taking the three points, Connacht went for the jugular, and a clever kick from O'Connor found O'Halloran who gathered well and crossed the whitewash. O'Connor deservedly levelled the scores with his conversion.
With Eric Elwood's troops sniffing their first win since in the league since September, ill-discipline from second row Eoghan Grace allowed Duncan Weir - on for Jackson - to kick the Warriors back in front with 12 minutes remaining.
Only a remarkable tackle in the corner from John Muldoon prevented Peter Murchie sealing the game for Sean Lineen's side, and that intervention was to prove key as O'Connor secured a hard-fought draw for his side.
|Connacht Rugby Score Card|
|Glasgow Warriors Score Card|