Glasgow fought back and after 15 minutes of pressure Dan Parks scored after a good Break from Graeme Morrison. The conversion from Parks made it 21-15.
Glasgow probed for an equaliser before a breakout thanks to Warwick and then Slemen led to a penalty which the former converted. Connacht had regained the initiative and from a lineout and drive they drove over with some power. John Muldoon got the touchdown and the bonus point.
Tries from Matt Lacey in the 5th minute and Paul Warwick in the 37th minute had put Connacht six points ahead at half time in their Celtic League opener. It could have been a bigger lead but for a late second half score from Kenny Logan for Glasgow on 40 minutes.
Connacht had looked the better side overall but three times Glasgow failed to take advantage of good scoring opportunities in Connacht's 22. That try for Glasgow really gave them a new lease of life though after they had looked in real trouble when Paul Warwick split their defence form ten metres with three minutes left.
Connacht: 15. David Slemen; 14. Ted Robinson, 13. Matt Mostyn, 12. James Downey, 11. Conor McPhillips; 10. Paul Warwick, 9. Tom Tierney; 1. Adrian Clarke, 2. John Fogarty, 3. Peter Bracken, 4. Christian Short, 5. Andrew Farley, 6. John Muldoon, 7. Matt Lacey, 8. John O'Sullivan.
Replacements: 16.Henry Bourke, 17. Stephen Knoop, 18. Michael Carroll, 19. Paul Neville, 20. Conor O'Loughlin 21. Eric Elwood, 22. Niall O'Brien
Glasgow: 15 Rory Kerr; 14 Sean Lamont, 13 Andrew Henderson, 12 Scott Barrow, 11 Kenny Logan; 10 Dan Parks, 9 Sam Pinder, 1 Kevin Tkachuk 2 Gordon Bulloch, 3 Lee Harrison, 4 Joe Beardshaw, 5 Andrew Hall, 6 Paul Dearlove, 7 Donnie Macfadyen, 8 Jon Petrie (captain). Replacements: 16 Scott Lawson, 17 Andrew Kelly, 18 Dan Turner, 19 Andrew Wilson, 20 Graeme Beveridge, 21 Calvin Howarth, 22 Graeme Morrison.