He said after the 29-16 success: "The boys really stepped to the plate today. It was a very physical encounter and we were very strong in defence.
"It was a ding-dong battle and neither side were too far ahead in any area of the game. Up front the boys probably edged it and put the squeeze on the Saracens especially in their own 22."
Shane Geraghty was a late call-up to the starting line up and he scored the decisive try with a crucial interception.
Smith continued: "Shane only knew he was starting at 11am as Jonathan Joseph stubbed his toe in training. I am especially pleased for Shane as he has skippered the team well in the past few weeks and responded brilliantly to the call up, scoring the defining try."
Friday night's win for fellow strugglers Sale had put increased pressure on Irish but they responded magnificently to win the game, and Smith said: "I didn't watch last night's game as I couldn't - it would have given me too many butterflies. I checked the result but didn't refer to it once when speaking to the boys.
"We are looking ahead and are aiming at 16 points from our last eight games with five of those at home. However we must not look too far ahead."
Smith's Saracens counterpart, Mark McCall, accepted his side were second best, and he explained: "We were well beaten today with full credit to Irish, as they put us under a lot of pressure.
"I am very disappointed that when we were chasing the game there was a lot of poor decision making. We had a good team out and didn't lack effort but there were too many mistakes and poor choices.
"There were few positives to take from the game, David Strettle was very busy and scored our try and our young loosehead Nick Auterac performed with credit but little else. However we don't have many bad performances and have a good record with 19 wins from 24 games this season."