The Welsh side had been 11-8 in front at half-time following early penalties from Jarrod Evans and Aled Summerhill’s try.
However, Leicester had kept themselves in touch with a try from Jordan Coglan, his first for the club, and Hardwick’s penalty.
The visitors drew level on the hour through another successful kick from fly-half Hardwick.
Tempers boiled over during the closing stages as players from both sides were involved in a melee.
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The Season 5 – Episode 3
A changing room reminder about team values and expectations encourages the squad to train with consistency but mistakes start to creep in.
As the clock ticked down, Nephi Leatigaga got over the ball at a breakdown, and Hardwick kept his composure to convert a match-winning penalty and secure a second Pool 5 win for the Tigers.
Earlier, Wasps had got their own European campaign up and running with a 32-14 win over Agen at the Ricoh Arena.
The home side, beaten away in their Pool 3 opener to Bordeaux Begles last weekend, had trailed to a converted try from Loris Tolot.
Ashley Johnson went over from a maul before Jacob Umaga’s kicking extended the lead to 13-7 at the break.
London Irish ran in seven tries to beat Bayonne 45-31 in their Pool 2 match at the Madejski Stadium.
The Exiles, beaten 20-16 against Scarlets last time out, scored five tries in the first half – during which the French club also crossed four times to go into the break at 31-24.
Converted tries for Saia Fainga’a and Ben Meehan gave the hosts some breathing space. A late score for former Harlequins number eight Mat Luamanu, his second of the match, secured Bayonne a bonus point.
Stand-off Paddy Jackson kicked five conversions, including one from his own try after 25 minutes, to finish with a personal haul of 15 points.
Saturday’s early kick-off in Pool 1 saw Worcester Warriors beaten 17-9 away at Castres.
Elsewhere, in Pool 4, Stade Francais picked up a much-needed 13-12 win at Zebre.
The Italians, though, made a late fight back with a penalty try setting up a tense finale.
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