Edinburgh had Tim Visser to thank as they continued their good run of form against Ulster with a 17-13 Heineken Cup win at Murrayfield on Saturday.
With just over 10 minutes remaining Visser went over for an unconverted try before Godman sealed victory with a last-minute penalty.
Edinburgh had gone down to Stade Francais in their Heineken Cup opener while Ulster had beaten Bath and the spirits of the Edinburgh fans among the crowd of 5747 will have sunk before kick-off with the news that multi-capped, multi-points kicking full back Chris Paterson was out with a viral infection replaced by Steve Jones.
However, his kicking replacement Godman gave his side the best possible start by kicking home a penalty in the first minute from just inside the Ulster half.
Ulster adopted a conservative approach with number 10 Ian Humphreys repeatedly stabbing the ball into good positions.
Edinburgh tried to raise the pace but to little effect. However, in the 20th minute a drive from a line out brought a penalty on the Ulster 22 that Godman kicked home.
However, from the re-start the whole face of the game changed. Ulster winger Timoci Nagusa broke from the edge of the ruck and the Edinburgh defence was left looking at one another as he cut inside to the posts from where Humphreys converted.
This put Edinburgh into their shell and things only got worse. After Godman had missed a long penalty Humphreys kicked one to open the points gap to 10-6.
Edinburgh at last got some zip into their play as the interval beckoned but Godman had a drop goal charged down and then missed his second penalty.
In control Ulster began the second half with a slick penalty to touch, line out drive, followed by a penalty against Edinburgh kicked by Humphreys.
Again Edinburgh raised the pace and following a penetrative run by Roddy Grant, Nick De Luca was held up on the line.
With the game now moving into the last quarter and the home substitutes adding impetus Edinburgh became established in the visitors' half.
The pressure brought a penalty when the scrum went down. Godman struck it home in the 67 minutes and even better was to follow two minutes when Godman set De Luca free.
He carved an opening before releasing Dutch winger Tim Visser who got over in the corner for an unconverted try. Godman confirmed the win with a last-minute penalty concluding constant Edinburgh pressure.
Pen: Godman 4
Edinburgh: 15 Chris Paterson, 14 Mark Robertson, 13 Nick De Luca, 12 John Houston, 11 Tim Visser, 10 Phil Godman, 9 Mike Blair (capt), 8 Allister Hogg, 7 Roddy Grant, 6 Alan MacDonald, 5 Scott MacLeod, 4 Steve Turnbull, 3 Geoff Cross, 2 Ross Ford, 1 Kyle Traynor.
Replacements: 16 Andrew Kelly, 17 Allan Jacobsen, 18 David Young, 19 Craig Hamilton, 20 Scott Newlands, 21 Greig Laidlaw, 22 Ben Cairns, 23 Steve Jones.
Ulster: 15 Clinton Schifcofske, 14 Timoci Nagusa, 13 Darren Cave, 12 Paddy Wallace (capt), 11 Andrew Trimble, 10 Ian Humphreys, 9 Isaac Boss, 8 Robbie Diack, 7 Willie Faloon, 6 Stephen Ferris, 5 Ryan Caldwell, 4 Ed O'Donoghue, 3 Brendon Botha, 2 Andy Kyriacou, 1 Tom Court.
Replacements: 16 Nigel Brady, 17 Bryan Young, 18 Declan Fitzpatrick, 19 Dan Tuohy, 20 Chris Henry, 21 Paul Marshall, 22 Ian Whitten, 23 Simon Danielli.
Referee: David Pearson (England)
Assistant referees: David Rose(England), Paul Dix (England)
Television match official: Graham Hughes (England)
Assessor: Gabriele Villari (Italy)